Second-Order Predation and Pleistocene Extinctions: By Elin
Transcript of Second-Order Predation and Pleistocene Extinctions: By Elin
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Second-Order Predation and Pleistocene Extinctions:A System Dynamics Model
By
Elin Whitney-Smith
B.A. June 1975, Rutgers University, New Brunswick, NJ
M.S. June 1985, San Jose State University, San Jose, CA
Ph.D. February 1991, Old Dominion University, Norfolk, VA
A Dissertation submitted to
The Faculty of
Columbian School of Arts and Sciences
of the George Washington University in partial satisfaction
of the requirements for the degree of Doctor of Philosophy
May 20, 2001
Dissertation directed by
Henry Merchant, Ph.D.
Associate Professor of Biology
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For
Christoph Berendes
who has never lost faith
seldom lost patience
and continues to be there.
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Abstract
At the end of the Pleistocene, there were significant climate changes and, following the
appearance of Homo Sapiens on each major continent, significant megafaunal
extinctions.
The leading extinction theories, climate change and overkill, are inadequate. Neither
explains why: (1) browsers, mixed feeders and non-ruminant grazer species suffered
most, while ruminant grazers generally survived, (2) many surviving mammal species
were sharply diminished in size; and (3) vegetative environments shifted from plaid to
striped (Guthrie, 1980.)
Nor do climate change theories explain why mammoths and other megaherbivores
survived changes of similar magnitude.
Although flawed, the simple overkill hypothesis does link the extinctions and the
arrival of H. sapiens. Mosimann & Martin(1975) and Whittington & Dyke( 1984)
quantitatively model the impact of H. Sapiens hunting on prey. However, they omit the
reciprocal impact of prey decline on H. Sapiens; standard predator-prey models, which
include this effect, demonstrate that predators cannot hunt their prey to extinction
without themselves succumbing to starvation.
I propose the Second-Order Predation Hypothesis , a “boom/bust” scenario: upon
entering the New World, H. sapiens reduced predator populations, generating a
megaherbivore boom, then over-consumption of trees and grass, and, finally,
environmental exhaustion and the extinctions.
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The systems dynamic model developed in this work (available in the CDROM
attached or from http://quaternary.net/extinct2000/) specifies interrelationships between
high and low quality grass, small and large trees, browsers, mixed feeders, ruminant
grazers and non-ruminant grazers, carnivores, and H. sapiens driven by three inputs: H.
sapiens in-migration, H. sapiens predator kill rates, and H. sapiens food requirements It
permits comparison of the two hypotheses, through the setting of H. sapiens predator kill
rates. For low levels of the inputs, no extinctions occur. For certain reasonable values
of the inputs, model behavior consistent with Second-Order-Predation: carnivore killing
generates herbivore overpopulation, then habitat destruction, and ultimately differential
extinction of herbivores. Without predator killing, extinctions occur only at unreasonable
levels of in-migration. Thus, Second-Order-Predation appears to provide a better
explanation.
Further, the boom-bust cycles suggest we “over-interpret” the fossil record when we
infer that the populations decreased steadily, monotonically to extinction.
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Contents -- Click to Access Page
Abstract .......................................................................................................... iii
Contents ......................................................................................................... v
Tables ............................................................................................................ xi
Figures..........................................................................................................xii
Chapter I: Introduction and Literature Review............................................... 1
Introduction ..................................................................................................................... 1
Characteristics of the Pleistocene–Holocene Transition and Its Extinctions.............. 3
Hypotheses Regarding the Cause of the Pleistocene-Holocene Extinctions............. 10
Criteria for New Hypotheses of Extinction............................................................... 28
An Alternative Hypothesis of Extinction...................................................................... 32
The Argument for Second-Order Predation .............................................................. 32
A Proposed Scenario of Pleistocene Extinctions Due to Second-Order Predation... 38
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Chapter II: A Method for Testing Hypotheses of Pleistocene Extinctions in
the New World .............................................................................................. 45
Introduction ................................................................................................................... 45
The Modeling Process............................................................................................... 46
General Conventions and Definition of Terms ......................................................... 47
General Overview of the Model................................................................................ 57
Criteria for Success ................................................................................................... 58
Base Model: Dynamic Equilibrium – Step 1 ................................................................ 59
Overview ................................................................................................................... 59
Model Diagram ......................................................................................................... 59
Conventions, Definitions and Equations................................................................... 62
Graph of the Base Model – Step 1 ............................................................................ 68
Second Predator: Overkill – Step 2a ............................................................................. 72
Overview ................................................................................................................... 72
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Model Diagram for Step 2a....................................................................................... 72
Conventions, Definitions, and Equations.................................................................. 74
Results Second Predator (Overkill) – Step 2a............................................................... 76
Graph of the Model – Step 2a ................................................................................... 76
Second-Order Predation – Step 2b ................................................................................ 78
Overview ................................................................................................................... 78
Model Diagram ......................................................................................................... 78
Conventions, Definitions, and Equations.................................................................. 78
Results Second-Order Predation – Step 2b ................................................................... 80
Graph of Model – Step 2b ......................................................................................... 80
Step 3 – Three Herbivores – (Browsers, Grazers and Mixed Feeders)......................... 83
Overview ................................................................................................................... 83
Diagram of the Model ............................................................................................... 83
Conventions, Definitions, and Equations.................................................................. 85
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Plants ......................................................................................................................... 89
Herbivores ............................................................................................................... 103
Results of Step 3: Three-Herbivore Model ................................................................. 129
Graph of the Model ................................................................................................. 129
Step 4 – Four Herbivores (Browsers, Ruminant Grazers, Non-ruminant Grazers and
Mixed Feeders)............................................................................................................ 136
Overview ................................................................................................................. 136
Diagram of the Model ............................................................................................. 136
Conventions and definition of terms used in Step 4................................................ 136
Results of the– Four-Herbivore Model ....................................................................... 163
Graph of the Model ................................................................................................. 163
Chapter III: Testing and Validity................................................................ 171
Introduction ................................................................................................................. 171
Tests for Suitability of Structure ................................................................................. 172
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Dimensional Consistency........................................................................................ 172
Extreme Conditions................................................................................................. 173
Boundary Adequacy................................................................................................ 179
Tests for Suitability of Model Behavior...................................................................... 181
Parameter sensitivity ............................................................................................... 181
Structural Sensitivity............................................................................................... 183
Tests for the Consistency of the Model with the Real system .................................... 183
Face Validity ........................................................................................................... 183
Parameter Values..................................................................................................... 184
Replication of Reference Modes ............................................................................. 185
Surprise behavior..................................................................................................... 186
Additional Characteristics Contributing to Model Utility and Effectiveness ............. 186
Appropriateness of Structure:.................................................................................. 186
Counterintuitive Behavior: ...................................................................................... 187
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Generation of Insights ............................................................................................. 188
Chapter IV: Conclusions and Significance................................................. 189
Introduction ................................................................................................................. 189
Conclusions ................................................................................................................. 189
Discussion ................................................................................................................... 190
Climate and Vegetation........................................................................................... 190
Animals ................................................................................................................... 192
Archaeological Evidence......................................................................................... 194
Implications for Further Research............................................................................... 196
Significance of the Model for Science and Research.................................................. 197
A Broader Significance for the Modern World........................................................... 200
Appendix A: Equations............................................................................... 201
Appendix B - Summary Graphs.................................................................. 218
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Appendix C – Model and Runtime Software on CD Rom......................... 222
Bibliography................................................................................................ 223
Tables
Table 1: Variables and starting values of the Mosimann and Martin and the Whittington
and Dyke models....................................................................................................... 19
Table 2: Features of the Pleistocene-Holocene transition................................................. 31
Table 3: Values taken directly from Whittington and Dyke (1984) ................................. 56
Table 4: Modified values based on Whittington and Dyke (1984) ................................... 56
Table – 3 – Comparison of Second Predator (Overkill) and Second-Order Predation
Ending Values ........................................................................................................... 82
Table – 4 – Array Illustration.......................................................................................... 139
Table 5 – Validity matrix based on Richardson and Pugh (1981) ................................. 172
Table – 6. – Carnivore population reduction................................................................... 183
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Figures
Fig. 1. Prorated rates of mammalian extinction (after Webb, 1989)................................... 2
Fig. 2 Correlation of the strata of the Pleistocene – Holocene transition in North America
(adapted from Haynes, 1984) ...................................................................................... 9
Fig. 3. The march of extinction (after Martin, 1984): ....................................................... 16
Fig. 4 Oscillation of predator and prey populations.......................................................... 24
Fig. 5 The path of extinction held by various hypotheses................................................. 34
Fig. 6. Effect of the arrival of a new predator on the populations of North American prey
and predators ............................................................................................................. 40
Fig. 7. Role of trees and grass in climate change:............................................................. 44
Fig. 8. Dynamic equilibrium reference mode ................................................................... 49
Fig. 9. Second-predator (overkill) reference mode ........................................................... 50
Fig. 10. Second-order predation reference mode .............................................................. 51
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Fig. 11. Illustration of the step function ............................................................................ 53
Fig. 12. Illustration of the pulse function .......................................................................... 54
Fig. 13. Base model diagram............................................................................................. 60
Fig. 14. Hunting function .................................................................................................. 66
Fig. 15. Graph of the base model ...................................................................................... 69
Fig. 16. Pulse outflow from plants, herbivores and carnivores......................................... 70
Fig. 17. Second predator (overkill) model diagrams......................................................... 73
Fig. 18. Graph of the second predator (overkill) mode..................................................... 77
Fig. 19. Second-order predation diagram.......................................................................... 79
Fig. 20. Graph of the second order predation model......................................................... 81
Fig. 21. Continent, trees and grass, diagram. Three herbivore model. ............................. 84
Fig. 22. Herbivores. – browsers, grazers, and mixed feeders diagram. Three herbivore
model......................................................................................................................... 86
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Fig. 23. Density diagram. Three herbivore model. ........................................................... 87
Fig. 24. Carnivores, Hsapiens diagram. Three herbivore model....................................... 88
Fig. 25. WoodMix function. Three herbivore model. ....................................................... 90
Fig. 26. MixedFeeder efficiency. Three herbivore model................................................. 95
Fig. 27. GzEffGr. Actual efficiency of Grazers given the amount of grass available in the
system Three herbivore model. ............................................................................... 100
Fig. 28. Effect of Browser density, as Browser density declines Browser birth function
drops toward zero. Three herbivore model. ............................................................ 105
Fig. 29. The rate at which Carnivores kill Browsers. Three herbivore model. ............... 107
Fig. 30. The rate at which Hsapiens kill Browsers. Three herbivore model. .................. 109
Fig. 31. Actual efficiency of Grazers given the amount of grass available in the system.
Three herbivore model. ........................................................................................... 111
Fig. 32. The birth rate of Grazers. Three herbivore model ............................................. 112
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Fig. 33. Effect of Grazer density, as Grazer density declines Grazer birth function drops
toward zero. Three herbivore model ....................................................................... 114
Fig. 34. The rate at which Carnivores kill Grazers. Three herbivore model................... 115
Fig. 35. The rate at which Hsapiens kill Grazers. Three herbivore model...................... 117
Fig. 36. The birth rate of MixedFeeders. Three herbivore model. .................................. 120
Fig. 37. Effect of MixedFeeder density, as MixedFeeder density declines MixedFeeder
birth function drops toward zero. Three herbivore model. ..................................... 121
Fig. 38. The rate at which Carnivores kill MixedFeeders. Three herbivore model. ....... 123
Fig. 39. The rate at which Hsapiens kill MixedFeeders. Three herbivore model. .......... 125
Fig. 40. The death rate of MixedFeeders according to the amount of grass in the
environment relative to the amount needed. Three herbivore model...................... 127
Fig. 41. The rate Hsapiens hunts Carnivores relative to their density. Three herbivore
model....................................................................................................................... 128
Fig. 42. Equilibrium mode graph. Three herbivore model.............................................. 130
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Fig. 43. - Second predator (overkill) mode, aggregated view. Three herbivore model. . 131
Fig. 44. Second-order predation, aggregated view. Three herbivore model ................... 133
Fig. 45. Second-order predation, herbivores Three herbivore model ............................. 134
Fig. 46. Second-order predation, plants Three herbivore model..................................... 135
Fig. 47. Grass, diagram. Four-herbivore model. ............................................................. 137
Fig. 48. Grazers diagram. Four-herbivore model............................................................ 138
Fig. 49. Actual efficiency of ruminant grazers (Grazers[Ruminant]) given the amount of
grass available in the system. Four-herbivore model. ............................................. 142
Fig. 50. Actual efficiency of non–ruminant grazers(Grazers[NonRuminant]) given the
amount of grass available in the system. Four herbivore model............................. 144
Fig. 51. The birth rate of ruminants Grazers[Ruminant] Four herbivore model............. 147
Fig. 52. Effect of Grazer[Ruminant] density, as Grazers[Ruminant] density declines
Grazers[Ruminant] birth function drops toward zero. Four-herbivore model. ....... 149
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Fig. 53. The rate at which Carnivores kill ruminant grazers (Grazers[Ruminant]). Four
Herbivore Model. .................................................................................................... 151
Fig. 54. The rate at which Hsapiens kill ruminant grazers (Grazers[Ruminant]). Four
Herbivore Model. .................................................................................................... 153
Fig. 55. The birth rate of non-ruminants (Grazers[NonRuminant]). Four-herbivore model.
................................................................................................................................. 156
Fig. 56. Effect of non-ruminant density, as Grazers[NonRuminant] density declines
Grazers[NonRuminant] birth function drops toward zero. Four-herbivore model. 157
Fig. 57. The rate at which carnivores kill non-ruminant grazers (Grazers[NonRuminant]).
Four-herbivore model.............................................................................................. 159
Fig. 58. The rate at which Hsapiens kill non-ruminant grazers (Grazers[NonRuminant]).
Four-herbivore model.............................................................................................. 161
Fig. 59. Equilibrium mode graph. Four-herbivore model. .............................................. 164
Fig. 60. Second predator (Overkill), Hsapiens enters the New World. Four-herbivore
model....................................................................................................................... 165
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Fig. 61. Second-order predation, aggregated view. Four-herbivore model. ................... 167
Fig. 62. Second-order predation, herbivores. Four-herbivore model.............................. 168
Fig. 63. Second-order predation, plants. Four-herbivore model. .................................... 169
Fig. 64. A. Second-order predation comparative graphs of A. browsers and B.mixed
feeders C.Aggregate with AmtHsKillCrn=0.015. Four herbivore model ............... 170
Fig. 65. Comparative populations predicated on varying migration of H. sapiens over
time.......................................................................................................................... 174
Fig. 66. Comparative population sizes predicated on varying food needs of H. sapiens
over time.................................................................................................................. 175
Fig. 67. Comparative populations predicated on varying food needs of H. sapiens and an
absence of second-order predation over time.......................................................... 176
Fig. 68. Herbivore populations where AmtHsMIgrate is set at 100,000, FoodNeedHs is
set at 10, and AmtHsKillCrn=0.075........................................................................ 178
Fig. 69. Interface to the model: ....................................................................................... 199
Chapter I: Introduction and Literature Review
Introduction
The greatest mammalian extinction event of the last ten million years occurred in North
America at the end of the Pleistocene epoch (called the Wisconsin by geologists and the
Rancholabrean by mammalogists). During a thousand-year period, more than 40
mammalian genera disappeared, an extinction rate of 77 percent prorated throughout the
stratigraphic interval. Thirty-nine of the 40 genera were large mammals. In geologic
terms, this extinction, occurring in thousands rather than millions of years, was
extraordinarily fast. By way of contrast, the second fastest mammalian extinction took
place in the Late Hemiphillian. It extended over 1.5 million years and involved 62
genera, 35 of which were large mammals (Webb, 1984). Prorated rates of mammalian
extinction during various periods are shown in Figure 1.
Throughout time, the origin and extinction of species and genera have been part and
parcel of how evolution happens. As Raup (1992) states, “most species [that once
existed] are [now] extinct.” He discusses the five massive extinction events of the
Ordovician, Devonian, Permian, Triassic, and Cretaceous periods. An average of 65
percent of all species disappeared during these periods. Following hydrologists’ use of
extreme value statistics, Raup has developed a “kill curve,” which allows scientists to
make intelligent guesses about the likely time span of extinction events of a given
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Fig. 1. Prorated rates of mammalian extinction (after Webb, 1989)
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magnitude. Thus, it can be determined that during the Cretaceous-Tertiary (K-T) periods,
the extinction of virtually all plants and animals, on land and in the sea, from dinosaurs to
plankton, took place over a hundred million years. At the other end of the scale is a low
level of background extinction that occurs more or less all the time, although it appears to
take place over a long time frame when compared to the relatively short span of human
life. Few of these extinctions are considered complete, because replacements usually
occur. (Replacement happens when one species of animal disappears and another, similar
species takes its place, thereby utilizing the same or a similar niche.)
Characteristics of the Pleistocene–Holocene Transition and Its Extinctions
Late Pleistocene extinctions fall somewhere between the two extremes described above.
They were nowhere near the magnitude of the “big five” and did not have the same
characteristics. They involved primarily large mammals and birds, rather than all life
forms as in the K-T period. On the other hand, the Pleistocene extinctions, such as the
disappearance of the saber tooth cat, the mammoth, and the mastodon, were without
replacements. Others were geographically limited; the camel and the horse, for example,
became extinct in the New World, but survived in parts of Europe, Africa, and Asia.
Gingerich (1984) suggests that the rate of replacement may be more significant than
the rate of extinction. For example, 56 percent of the large mammals that disappeared in
the Rancholabrean were not replaced by new species. That is, more than half of the
mammalian extinctions did not result in one species evolving into another with similar
niche demands (e.g., Bison antiquus into Bison bison). Instead, there were many species
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that disappeared completely (Webb, 1984; Gingerich, 1989). As a result, the
Rancholabrean extinctions were far in excess of those low-level background extinctions
that take place continually among all life forms.
It is possible that the extinctions of the Rancholabrean were simply the result of
random events. However, the magnitude and rate of extinction, the high proportion of
non-replacement, and the bias toward the extinction of large mammals strongly suggest
that they were caused by factors other than chance.
Ecosystem changes have been identified and described as a possible cause of the
extinctions of this period. The first such change was a general global warming, which
resulted in the disappearance of the Wisconsin ice sheet. In North America during the
height of glaciation, a sheet of ice that averaged more than a mile in thickness spread out
from Hudson Bay to bury all of eastern Canada, New England, and much of the Midwest.
A second ice sheet spread out from the Canadian Rockies and other highlands in western
North America to cover parts of Alaska, all of western Canada, and portions of Idaho,
Montana, and Washington. The final extent of the ice sheets’ edges and their subsequent
retreat can be traced in moraines (sedimentary deposits that accumulate at the terminus of
glaciers). In addition to these geomorphological signposts, evidence of planetary
warming comes from changes in the ratio of oxygen isotopes found in deep–sea cores
(Broecker & Van Donk, 1970; CLIMAP, 1976; Imbrie, 1985). These independent lines of
investigation are consistent in their demonstration of an increase in temperature of
roughly 6o Celsius. It is generally held that astronomical cycles are the mechanisms
forcing the alternation of glacials and interglacials (Imbrie & Imbrie, 1986).
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Some investigators have identified a second ecological change that took place at the
end of the Pleistocene, namely a decrease in moisture and a resultant increase in the
continentality of the climate. In North America, in comparison with other glacial-
interglacial transitions, the summers became hotter and the winters colder. Conversely,
Taylor (1965), working on mollusks found in the mid-continent, deduced that during
periods in the early part of the Pleistocene, winters were milder and frost-free, while the
summers were cooler. Further evidence for this change in continentality was a change in
the pattern of species association (Guthrie, 1968, 1980, 1990; Hoffman & Jones, 1970).
Plants (Martin & Mehringer, 1965; Davis, 1976; Delcourt & Delcourt, 1987), insects
(Ashworth 1977, 1980; Coope, 1967, 1977) and fauna (Kurten & Anderson, 1980;
Russell et al, 1984) that had lived together throughout most of the Pleistocene and other
temperate periods, were not found to be living together in Holocene environments. This
suggests that some species that once were able to live in the earlier, more temperate
climate, eventually found the summers too hot, while other found the winters too cold
As a result of these climate transformations, the pattern of vegetation changed. The
earlier, more heterogeneous, patchy environment was transformed into one comprised of
more specialized vegetation zones, i.e., grasslands in the center of the continent and
forests on the continental edges. Guthrie (1980) has described this change as one from
“plaid” to “striped” environments. The geographic region that today is associated with
prairie, or grassland, was more of a mixed woodland-parkland, a cross between a
temperate savanna and an open-canopy forest. Areas that today are closed-canopy forests
were once open, with many grassy places (Hopkins et al, 1967; Bryson et al, 1970;
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Hibbard, 1970; Wendorf, 1970; Birks & West, 1973; Anderson, 1974; Morgan &
Morgan, 1979, 1981; Brumley, 1978; Graham & Lundelius, 1989). According to Guilday
(1989):
The deterioration and disintegration of the eastern and western segments of this
parkland were, in some respects, mirror images of one another. In the western
segment, the Great Plains tree cover disappeared almost completely except for
scattered firebreak ridges and along river courses where corridor woodlands persisted.
In the eastern segment, as closed-canopy deciduous forest evolved, grasslands
became restricted primarily to river valley corridors. (p. 225)
In Alaska, sediments have yielded insect remains (Matthews, 1979) and pollen and
plant remains (Davis, 1976) that are associated with well-drained soil. They suggest a
parkland of low, grassy vegetation and scattered tree cover, rather than today’s tundra,
which is typified by treeless, poorly-drained, cold soils. The remains of a large and
diverse complement of animals, dominated by mammoth, bison, horse, and their
predators, lived in this parkland. Guthrie (1982) has named it the Mammoth Steppe
biome (Pruitt, 1960; Fuller & Bayrock, 1965; Flerov, 1967; Hoffmann & Taber, 1967;
Pewe & Hopkins, 1967; ; Hopkins et al, 1967;Repenning, 1967; Sainsbury, 1967;
Frenzel, 1968 Ritchie & Hare, 1971; Kurten, 1972, 1988; Yurtsev, 1972; Sorenson, 1977;
Batzli et al, 1980; Calef, 1984; Harrington, 1984; Guthrie, 1989).
Because climate and vegetation changes are strongly supported by the evidence
described above, it is highly probable that important changes in consumer populations
also occurred. Evidence from archaeological and paleontological sites shows a change in
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the spatial distribution of animal associations in the temperate zone. Many species of
animals that today are allopatric (occurring in disjunct geographic areas) were sympatric
(occurring in overlapping areas) during most of the Pleistocene. The literature from many
different fields is full of references to “defunct species associations” (Slaughter 1967),
“communities without modern or extant counterparts” (Matthews 1979), and
“disharmonious species associations” (Graham, 1976; Graham & Lundelius, 1989).
According to Guilday (1989):
. . .the broad belt of ecologically diverse, predominantly coniferous parkland that
extended from at least Wyoming east to the Atlantic Coastal Plain. . . disintegrated as
a biological unit within a relatively short period of time. . . its component species
either becoming extinct or regrouping themselves into assemblages that continued to
polarize to the present day. . . Neither of these corridors, wooded in the Plains,
grassed in the East, was extensive enough to support more than a few large mammals
on a sustained basis. (p. 225)
In addition to changes in the distribution of species associations, the Pleistocene–
Holocene transition was notable for its uneven impact on different groups of ungulate
mammals. A survey of extinct vs. extant animals (Anderson, 1989;Guthrie,1989)
indicates that the North American ungulate fauna of the Pleistocene was generally larger,
with comparatively more monogastrics, than those of the Holocene. According to Owen-
Smith (1992), the incidence of generic extinction correlates positively with body size. All
megaherbivore genera over 1,000 kilograms disappeared, compared with 76 percent of
genera in the 100 to 1,000 kilogram range, and 41 percent of genera between 5 and 100
kilograms.
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The final major change in the ecosystems of the New World during the late
Pleistocene was the arrival of H. sapiens. Haynes (1984) has done a survey of well–dated
(14C) and well–stratified sites in the United States. From this survey, he has extrapolated
a simplified pattern of the stratigraphy of the Pleistocene-Holocene. In each of the sites
surveyed, he has identified a turning point, from degradation (loss of material due to
glacial outflow) to aggradation (deposition of material due to sedimentation). He uses this
switch as a benchmark that can be traced across the country, tying together sites in
various locations. In all of these sites, evidence of megafauna is found below the
benchmark, with no evidence of artifacts. Above the benchmark, Paleo–Indian artifacts
occur for the first time in association with remains of megafauna. In sites that give
evidence of human occupation, the sequence is 1) Paleo–Indian artifacts with remains of
megafauna; 2) Paleo–Indian artifacts in association with transitional fauna remains; 3)
archaic artifacts, and then ceramics, both in association with the remains of modern fauna
(Figure 2). Thus, the picture drawn by Haynes’s survey shows that megafauna remains
exist at the lower levels , but became scarcer up through the stratigraphic sections, and
disappear entirely during the transition to the Holocene.
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Fig. 2 Correlation of the strata of the Pleistocene – Holocene transition in North America
(adapted from Haynes, 1984)
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Hypotheses Regarding the Cause of the Pleistocene-Holocene Extinctions
Since the timing of the Pleistocene-Holocene extinctions in North America correlates
roughly with climate change and the appearance of H. sapiens, it is not surprising that
they have been linked causally. This linkage has been expressed in two major categories
of hypotheses: those related to climate change and those associated with hunting by H.
sapiens.
Climate Change Hypotheses
When scientists first realized that there had been glacial and interglacial ages, and that
they were somehow associated with the prevalence or disappearance of certain species
and genera, they surmised that the termination of the Pleistocene ice age might be a
sufficient explanation for certain mammalian extinctions.
Increased Temperature
The most obvious change associated with the termination of an Ice Age is the increase in
temperature. Between 15kya and 10kya, a 6o Celsius increase in global temperatures
occurred at the climatic optimum. This was generally thought to present appropriate
conditions for an extinction event.
According to this hypothesis, a temperature increase sufficient to melt the Wisconsin
ice sheet also could have provided sufficient thermal stress to cold-adapted mammals to
cause them to die. The heavy fur of these mammals, which functioned to conserve body
heat in the glacial cold, might have impeded the dumping of excess heat into the warmer
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environment, so that they would have died of heat exhaustion. Large mammals, with their
reduced surface-area-to-body ratio, would have fared worse than small mammals.
Shortcomings of the Temperature Hypothesis
Perhaps the strongest argument against the temperature hypothesis is that since it was
first proposed it has become evident that today’s annual mean temperature is no higher
than that of previous interglacials (Andersen, 1973; Birks, 1973; Davis, 1976; Ashworth,
1980; Birks & Birks, 1980; Bradely, 1985). Because the large mammals survived similar
temperature increases in previous interglacials, warmer temperature alone is not a
sufficient explanation for the extinction of the Pleistocene megafauna. Furthermore, cold-
adapted animals, such as polar bears, are able to survive the warm summers in zoos
located in temperate zones.
Increased Continentality of Climate
The increase in continentality (hotter summers and colder winters) is also cited as either a
direct or indirect climate-related cause of extinction (Bryson et. al, 1970; Graham &
Lundelius, 1989; King & Saunders, 1989). Axelrod (1967) and Slaughter (1967) argue
that along with colder winters and hotter summers, the late Pleistocene experienced less
rainfall, which was also less predictable.
According to the supporters of this group of hypotheses, the changes in the amount
and/or distribution of rainfall could have affected the survival of the Pleistocene
mammalian megafauna by changing the amount and kind of vegetation that served as the
basis for the energy/nutrition relationships within the ecosystem. Graham & Lundelius
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(1989), Guthrie (1989), and Guilday (1989) attribute the extinctions to changes in the
floral environment. Graham & Lundelius (1989) say that because of the change in
continentality and the ensuing change from mixed woodland-parkland to separate prairie
and woodland environments (see above), there was a change in the kinds of food
available. They suggest that herbivores were unable to find the plants with which they
had co–evolved. As a result, they fell prey to the anti–herbivory toxins of the plants they
were able to find.
In a related vein, Hoppe (1978) and Guthrie (1980, 1989) argue that the extinction of
large herbivores and the dwarfing of many others was due to changes in the length of the
growing season. Based on their observations of modern fauna, they surmise that large
ruminants, such as bison, fared better than monogastrics, such as horses and elephants.
The large ruminants may have succeeded because they were able to extract more
nutrition from limited quantities of high-fiber food, and thus were better able to deal with
anti–herbivory toxins (Hoppe, 1978; Guthrie, 1980, 1989). However, this hypothesis has
been challenged by those who observe that increased continentality effected ecosystem
changes, which, in turn, resulted in an increased prevalence of grasses. McDonald (1981,
1989) suggests that the animals that became extinct actually should have prospered
during the shift from mixed woodland-parkland to prairie, because their primary food
source, grass, was increasing rather than decreasing (Birks & West, 1973; McDonald
1981, 1989).
Another hypothesis connecting megafaunal extinction to increased continentality
suggests that changes in rainfall restricted the amount of time favorable for reproduction.
13
Axelrod (1967) and Slaughter (1967) posit that large animals, with their longer, more
inflexible mating periods, often produced young at unfavorable seasons (i.e., when
sufficient food, water, or shelter was unavailable because of shifts in the growing season).
In this vein, Kilti (1989) suggests that the better survival rate of small animals may have
been due indirectly to the unpredictability of rainfall. He observes that the relationship
between the timing of gestation and the periods of available resources may have been
more important than the actual length of gestation per se. He says that small mammals,
with their shorter life cycles, shorter reproductive cycles, and shorter gestation periods,
were better able to adjust to the increased unpredictability of the climate, both as
individuals and as species. This better adjustment came about, in part, because their
reproductive efforts often coincided opportunistically with conditions favorable for
offspring survival. Even when their efforts did not coincide with good conditions, they
still risked less and lost fewer offspring than the large mammals, because they were better
able to repeat the reproductive effort later on, when circumstances once more favored
offspring survival.
Shortcomings of the Continentality Hypotheses
Critics of the continentality hypotheses have suggested a number of shortcomings. One
difficulty is that megaherbivores seem to have prospered in other continental climates.
For example, Holocene climates of North America are more continental today than they
were in the Pleistocene, but they are not more continental than the climate of Siberia
during the Pleistocene, in which megaherbivores were abundant (Flereov, 1967; Frenzel,
1968; McDonald, 1989).
14
Another argument against the continentality hypotheses addresses the presence of horses
in Holocene environments. The critics point out that although horses became extinct in
the New World, they were successfully reintroduced by the Spanish in the sixteenth
century. Today there are feral horses still dealing with environments similar to those in
the Holocene. They find a sufficient mix of food to avoid toxins, and they extract enough
nutrition from forage to reproduce effectively.
A third argument suggests that large mammals should have been able to migrate,
permanently or seasonally, if they found the temperature too extreme, the breeding
season too short, or the rainfall too sparse or unpredictable (Pennycuick, 1979). As a
modern-day example of this adaptation technique, African elephants migrate during
periods of drought to places where there is apt to be water (Wing and Buss, 1970).
Furthermore, season length is not simply a condition of temperature and humidity; it is
also a function of latitude. By moving south during the Pleistocene-Holocene transition,
Holarctic herbivores could have found areas with growing seasons more favorable for
finding food and breeding successfully.
Additionally, in studies of modern megaherbivores, Owen-Smith (1992) found that large
animals store more fat in their bodies than do medium-sized animals. Hence, increased
body mass should have encouraged an adaptation that compensated for extreme seasonal
fluctuations in food availability.
Finally, one of the soundest pieces of evidence against a purely climatic explanation is
the survival of members of extinct genera on isolated islands. According to Burney
15
In Europe, for instance, the last members of the elephant family survived climatic
warming not on the vast Eurasian land mass, but on small islands in the
Mediterranean—an even warmer climate. Radiocarbon dates suggest, for instance,
that dwarf elephants persisted on Tilos, a tiny island in the Aegean, until perhaps
4,000 to 7,000 years ago. (Burney, 1993)
Dwarf mammoths also survived on Wrangel Island in the Siberian Arctic until , until
4,000 to 7,000 years ago. (Vartanyan, 1993)
The theory that held the combined impact of hunting and fire caused the extinctions
now appears to be overly simplistic, according to the research of Burney and others on
Madagascar. The analysis of fossil pollen and charcoal in sediment cores from
throughout the island have shown that wildfires and vegetation changes were a normal
part of the environment for 35,000 years or more before the arrival of people. (Burney,
1993)
The Overkill Hypothesis: Hunting by H. sapiens
The overkill hypothesis suggests that humans hunted New World megaherbivores to
extinction. As a result, carnivores and scavengers that depended upon those animals
became extinct from lack of prey (Martin, 1963, 1967, 1984, 1986, 1988; Reeves, 1978;
Scott, 1984). This hypothesis is based on the observation that throughout the world,
extinctions have followed the emergence of H. sapiens. As shown in Figure 3, large
segments of fauna disappeared soon after humans entered the scene. Moreover, the
severity of extinction was greatest in areas where humans arrived relatively late, such as
16
Fig. 3. The march of extinction (after Martin, 1984):
A. loss of mammalian species relative to the immigration of H. sapiens; and B. extinct
fauna and archaeological evidence
A
B
17
the New World and Madagascar, and was least where humans arrived early, such as
Africa.
The Rationale Behind the Hypothesis
The rationale behind this hypothesis is that “naive” animals (ones who have no
experience with humans), are more easily killed than animals that evolve in the presence
of H. sapiens. Following this line of reasoning, the argument goes that because H.
sapiens originated in Africa, African animals had the most experience with H. sapiens
and presumably learned and evolved in response to the new hunting techniques that
threatened them. When H. sapiens entered other areas later in time, the animals they
encountered were naive about hunting and hence more vulnerable. There was a greater
gap between the ability of the hunter to kill and the ability of the animal to avoid being
killed. Flannery (1995) and Diamond (1984, 1997) have both given modern examples of
hunters coming within easy killing range of naive animals.
The rationale described above not only explains the timing of the disappearance of large
components of the fauna; it also explains why mammalian megafauna still exists in
Africa, but is depauperate in the rest of the world (Martin, 1966; Dawkins & Krebs, 1979;
Foley, 1984; Flannery, 1995).
In the New World, the presence of projectile points found imbedded in the bones of
extinct animals , as well as the presence of bones of extinct animals found in association
with archaeological sites, suggest that hunting by H. sapiens was directly responsible for
the extinction of the Pleistocene mammalian megafauna. Applying the overkill
18
hypothesis to this evidence, one can assume that evolution did not equip these animals to
withstand the onslaught of technologically-supported predation when humans finally
arrived on the scene. (Hester, 1967; Frison, 1974, 1978; Brumley, 1978).
Computer Simulations of the Overkill Hypothesis
Mosimann and Martin (1975) and Whittington and Dyke (1984) have designed computer
simulations of their “blitzkrieg” version of the overkill hypothesis (Martin 1974). In the
model designed by Mosimann and Martin (1975), human population is assumed to have
begun with 100 individuals around Edmonton, Canada. This population, with a densely
consolidated front of hunters, subsequently fanned out across the continent in an arc that
expanded geometrically (simulations were run at various population growth rates,
between 2.4 and 3.5 percent per year). The designers of this model likened the advance to
a blitzkrieg, or military invasion of great force and speed, in which the front remained
stationary only until all the megafauna in a given area were extinct. Then it would move
on at a rate of 20 miles per year. It was assumed that the hunters in the front were so
densely positioned that megafauna were not able to cross through the line. In fact, it was
assumed that the major factor causing extinction was the density of the front rather than
the overall density of the human population behind it. It was also assumed that the reason
for the lack of archaeological evidence was the speed with which the front advanced. In
the simulation, the front reached the coast in about 300 years and megafaunal extinction
occurred within three years of that time. Biologically, the model was based on
observations of the spread of snails, fish, and large herbivores into previously
uninhabited, but now habitable regions (Mosimann & Martin, 1975; Whittington & Dyke,
19
1984). Table 1 reports the variables and starting values of the Mosimann and Martin
model.
Table 1: Variables and starting values of the Mosimann and Martin and the
Whittington and Dyke models
Mosimann & Martin (1975) Whittington & Dyke (1984)Description of Variable
Source Source
Human population size
(individuals)
100 Arbitrary 200 Budyko, 1967, 1974
Human population growth rate
(percentage per year)
0.024 Birdsell, (1957);
Caughley, (1969)
0.0443 Birdsell, (1957)
Prey carrying capacity (a.u. *
per sq. mile)
25 Martin, (1973) 25 Mosimann and Martin,
(1975)
Prey biomass replacement rate
(a.u. * per sq. mile)
0.25 0.25 Mosimann and Martin,
(1975)
Human carrying capacity
(individuals per sq. mile)
1.295 Budyko, (1967, 1974)
Initial prey biomass (a.u. * per
sq. mile.)
25 25 Mosimann and
Martin, (1975)
Prey destruction rate (a.u.* per
person, per year)
3.862 Derived
* a.u. = animal units = 1K lb. of herbivore
The model created by Whittington and Dyke (1984) is based on the Mosimann and
Martin model (1975), but does not assume a front. For the most part, it contains the same
base values with some additions, as indicated in Table 1. In this model, the human growth
rate begins to decrease, thus doubling the time needed for population density to force the
20
periphery of the inhabited area to move at 20 miles a year. The baseline values include,
human carrying capacity, which is defined as 1.295 individuals per square mile, based on
the upper limit of estimates of human density in Europe at the end of the Upper
Paleolithic (Budyko, 1967, 1974, in Whittington & Dyke, 1984).
Whittington and Dyke’s (1984) prey destruction rate is 38.6 pounds of animal per pound
of H. sapiens per year. They derived the baseline value for the prey destruction rate by
running the model with the baseline values and varying the prey destruction rate until it
resulted in megafaunal extinction. They found that it represents 3,862 pounds of animal
killed per person annually, or, if we assume 100 lbs. average weight, 38.6 pounds of
animal per pound of H. sapiens per year.
Whittington and Dyke ran the model, employing various starting values and rates.
They found that extinction of megafauna will occur once the human population reaches
its carrying capacity, defined as “critical density” by Mosimann and Martin (1975), and
as “human carrying capacity” by Whittington and Dyke (1984). They found this to be
true even though the value of the prey-destruction is only 0.0001 a.u. above what the prey
biomass replacement rate will support. They state:
The only way to avoid megafaunal extinction is to reduce the human population
growth rate to zero before the [human] density reaches the threshold [carrying
capacity] . No matter how slowly the human population continues to grow while
hunting at its old rate, once the threshold is reached the megafaunal population will
fall quickly in density and size. This will occur even though the value of the prey
destruction rate is only 0.0001 a.u. (animal unit, 1 a.u. = 1,000 pounds of living prey)
above what the prey biomass replacement rate will support, an amount imperceptible
21
to a hunter. Even if humans switch to more readily available prey when the animal
biomass begins its crash, as long as they continue to occupy the entire continent and
kill megafaunal prey when they come upon it, the extinction process will continue.
(Whittington and Dyke, p. 463)
Shortcomings of the Overkill Hypothesis
Computer simulations add credence to the overkill hypothesis. However, all computer
models are based on the modeler’s assumptions. Some of the assumptions in question
here should receive further examination.
First, the models assume that a destruction rate greater than the replacement rate will
always lead to extinction. However, because human hunting rates are not driven by prey
replacement, it does not matter that humans cannot perceive the difference
Second, the derived hunting rate seems unrealistically large. There is simply no evidence
that H. sapiens needs more meat than non-human predators. Yet Whittington and Dyke’s
rate of 38.6 pounds of prey per pound of H. sapiens per year is almost double the needs
of a large felid, which requires 20 pounds of food per pound of cat per year (The Cat
House, 1996). It may be argued that H. sapiens is more wasteful than are cats. However,
to balance that, H. sapiens is an omnivore, who eats vegetables as well as meat,
The third assumption is that the carrying capacity for humans remains constant.
“Carrying capacity” is a term that expresses the concept that the resources of an area can
satisfy the needs of only a finite number of individuals. According to this concept, each
individual requires some constant fraction of the area to meet its resource needs. Thus,
the value of the carrying capacity can be expressed as the amount of area required to
22
support one individual, or more commonly, as the maximum number of individuals that
can be supported by a given area. Definition of the carrying capacity on an areal basis
assumes that the resources available to meet the needs of the consumers are everywhere
the same all of the time, and remain so throughout the entire simulation. This assumption
is appropriate under conditions in which the environment remains homogeneous and
unchanging, and one in which the constant consumption of resources is exactly balanced
by the constant replenishment of those same resources. Both the “critical mass” of the
Mosimann and Martin model (1975) and the “human carrying capacity” of the
Whittington and Dyke model (1984) express carrying capacity in these terms. But, in
fact, the many changes in abiotic and biotic features at the end of the Pleistocene are
themselves clear evidence that this assumption is unrealistic.
The introduction of hunter-gatherer H. sapiens into the ecosystem of North America
must have disrupted the existing predator–prey relations in the same ways that adding an
exotic predator disrupts a present–day balanced predator–prey system. Specifically, the
prey population decreases as it bears the impact of an additional predator. Therefore the
carrying capacity for predators decreases. Because the Pleistocene megafauna became
extinct, it is reasonable to assume that the prey populations could not replenish
themselves, and so the carrying capacity for the new predators (H. sapiens) had to have
declined. Yet the overkill hypothesis argues for a constant carrying capacity, based on a
steady rate of resource replenishment.
In formal models, it is generally assumed that predators and prey are linked in a mutually
causal loop (Caughley, 1970; May, 1973; Roff, 1975; Gluckenheimer et al, 1976; Hanby
23
& Bygott, 1979; Smuts, 1979; Hilborn & Sinclair, 1979; Cushing, 1984; Schaffer, 1988).
Thus, as prey populations increase, predators increase. As prey populations decrease, so
do predator populations. This makes it impossible for predators to kill off all their prey as
illustrated in Figure 4. In the real world, the kind of oscillations shown in the formal
model have been observed in simple ecosystems (Elton & Nicholson, 1942; Bulmer,
1974; Batzili et al, 1980). In complex ecosystems, however, as a prey species becomes
scarce, predators hunt prey that is more plentiful. This smoothes the oscillations and
makes it less likely that the prey will become extinct.
In the argument above, it is assumed that human and non- human predators behave
similarly – that they tend to switch prey as it becomes scarce, or that predator populations
will be reduced in response to a reduction in prey populations. Even so, Whittington and
Dyke (1984) imply that human predators are less likely than non- human predators to
switch prey.
At this point, it may be appropriate to argue that human predators are quite dissimilar
to non-human predators. I would suggest that humans are unlike non-human predators in
two respects. First, they are able to think economically; and second, they are omnivores.
For both of these reasons, humans are more likely to switch prey than non- human
predators. In support of the economic argument, Hawks and O’Connell (1994), in
observations of current-day hunter-gatherer groups, found that it is possible to predict
which foods will be utilized or discarded on the basis of frequency of resource encounter
and relative profitability (i.e., the rate of time spent on post–encounter pursuit, capture,
24
Fig. 4 Oscillation of predator and prey populations
25
and processing). Thus, as the frequency of encounter decreases, other resources will be
utilized. This suggests that human predators are likely to switch prey sooner than non-
human predators and are therefore less likely to overhunt their prey than non- human
predators.
Finally, animals that were not known to have been hunted by H. sapiens, such as the
Shasta ground sloth, became extinct at the same time as other Pleistocene mammals. The
overkill hypothesis does not specifically address this issue. In fact, by following the
reasoning behind the hypothesis, one would have to assume that because the ground sloth
became extinct, it must have been hunted by H. sapiens.
Combination Hypotheses
Diamond (1984) observes that extinctions in historic time have been brought about by a
variety of causes, ranging from simple overhunting, such as happened to the Great Auk
(Pinguinus impennis), through complex combinations of effects. He says of climate-
induced extinctions:
In considering the modern effects of climate, we had to content ourselves with
examining range contractions and local extinctions, because almost no modern cases
exist of total extinction due clearly to climate. (p.838)
By contrast, Diamond says of human-induced extinctions:
Modern man has proven versatile as an exterminator, with at least six major methods
long at his disposal (and a seventh, chemical pollution, recently added): overkill;
habitat destruction by logging, fire, induced browsing and grazing animals and
26
draining; introduction of predators; introduction of a competitor; introduction of
diseases; and extinctions secondary to other extinctions. All six modes were probably
effective in prehistoric extinctions as well. (p. 839)
This way of thinking about extinctions is useful. Diamond (1984) gives examples of
secondary extinctions, or “trophic cascades” (a succession of events based on nutritional
relationships) in history. One example is the extinction of ground-nesting birds on Barro
Colorado Island.
…insularization led to the loss of the largest predators (jaguar, puma, Harpy Eagle)
leading to a population explosion of smaller predators such as monkeys, pecaries,
coatimundis, and possums that served as their prey and that in turn rob bird nests.(p.
845)
In the same paper, Diamond also points out that just as there are a variety of
individual methods that have produced extinction, there are also cases in which a
combination of factors were at work.
The Heath Hen (Tympanuchus cupido cupido) was. . . shot by the thousands for food,
preyed upon by introduced cats, and afflicted with diseases of introduced poultry, all
the while its grassland habitat was being converted to farmland. By1830 it was
confined to the island of Martha’s Vineyard where its numbers rose to 20,000 by
1916. In that year its numbers were decimated by a fire in the summer, followed by a
harsh winter and the invasion of Goshawks. Cats, inbreeding and a disease introduced
with turkeys reduced its number to 13 in 1928, 2 in 1929, and in 1939 one, which
died in 1932. (p. 846)
The theory that held the combined impact of hunting and fire caused the extinctions
now appears to be overly simplistic, according to the research of Burney and others on
27
Madagascar. The analysis of fossil pollen and charcoal in sediment cores from
throughout the island have shown that wildfires and vegetation changes were a normal
part of the environment for 35,000 years or more before the arrival of people. (Burney,
1993)
Tim Flannery (1995) examines the ecology of the Australasian islands and describes a
variety of anthropogenic extinctions. In his work, he says that humans either adapted or
failed to adapt to the ecology of the area. On many of the Australasian islands, humans
hunted the fauna to extinction; then they came close to going extinct themselves. For
example, Flannery suggests that the success of Australian aborigines was due largely to
their use of “firestick farming.” He suggests that the use of fire was a cultural adaptation
necessitated by the extinction of Australia’s megaherbivores. He states:
After their extinction, fire became the main consumer of Australian vegetation.
Without human interference the fire pattern in Australia would probably have been
one of vast, periodic wildfires that ravaged huge areas of the continent. Indeed, this is
precisely the kind of fire regime that has emerged over much of the continent since
Aboriginal firestick farming ceased. . . It seems entirely possible that firestick
farming initially evolved as a response to the threat that the natural fire regime posed
to middle sized mammals. (p.240)
Australian megaherbivores, before their extinction, were what Owen-Smith (1992) calls
“keystone species.” By this he means that they served to maintain the balance of the
ecosystem. Once a keystone species is removed, the balance is broken and the ecosystem
becomes unstable until another equilibrium is found. Fire became the way aboriginal
people kept the forests from covering the environment and taking over grazing land. All
28
the early explorers of Australia record the use of fire by aboriginal people and describe
the open-woodland landscapes that the fires helped to sustain. Since the domination of
the continent by Europeans, aboriginal use of fire has been suppressed. This has resulted
in a transition from open woodlands to rainforest. In the absence of fire, vegetation has
filled in spaces that once were open (Flannery, 1995).
Burney suggests from the evidence from Madagascar that
…many of our sites show evidence for a combination of simultaneous changes,
including natural climate change, activities of the first human hunters, changes in fire
regime and vegetation structure, and the arrival of exotic species. I have come to refer
to the extinction event in Madagascar about 1,000 years ago as a “recipe for disaster.”
Instead of finding overwhelming evidence for the actions of a single cause in the
extinctions, these four factors, and perhaps others, seem to have been functioning
simultaneously …(Burney, 1993)
Criteria for New Hypotheses of Extinction
It seems eminently reasonable to think about extinctions as resulting from a variety of
anthropogenically-related factors, which caused a collapse of one ecosystem into another.
To understand these factors, a closer examination of the characteristics of an extinction
event may yield criteria to use in evaluating various hypotheses.
In one study, a condition of scarcity is suggested by the narrowness of growth rings on
mammoth tusks, which are wide when conditions are good, and narrow when conditions
are poor (Haynes, 1995, 1998). Dietary stress is also implicated as a cause of extinction
in an isotopic analysis of mammoth and mastodon remains (Koch, 1998). However,
29
neither climate change nor overkill by H. sapiens should have resulted in scarcity
conditions during the Pleistocene-Holocene transition. As herbivores were reduced in
numbers due to hunting, the competition between herbivores for food would have been
reduced. In fact, there would have been a net increase in available food. In addition, the
retreat of the ice sheet would have freed up more land for vegetation; overkill, by
eliminating some of the herbivores, would have resulted in more plants per remaining
animal.
Holocene fauna, in contrast to Pleistocene fauna, had a bias in favor of ruminants
(Graham, 1998; Guthrie, 1989; Koch, 1998). Ruminants extract energy and nutrients
more efficiently from their food than do monogastrics. Therefore, they need less food to
survive and to reproduce (Janis, 1975; Guthrie, 1989). The bias in favor of ruminants
suggests to this researcher a condition of scarcity of plant food.
These facts and their implications suggest that any new hypothesis should evaluate a
combination or factors and should address and explain 1) the extinction of horses in
North America; 2) the extinction of the ground sloth; 3) the bias in favor of ruminants;
and 4) the bias in favor of small mammal size.
The development of better simulation technologies presents an opportunity to develop an
alternative simulation model that takes into account these new observations. Such a
model may be able to eliminate the unrealistic assumptions in the existing models, which
were developed in support of the overkill hypothesis.
30
For an alternative hypothesis to be accepted, it must take into account all of the
features that are generally accepted as characteristic of the Pleistocene-Holocene
transition. These features are reported in summary form in Table 2. In addition, the
alternative hypothesis includes relevant information and understandings that come from
the ecological study of the interactions within and among present–day animal and plant
populations and communities. For example, Box A of Figure 5 presents the change in
mammalian megafauna populations as revealed by the fossil evidence. The fossil record
indicates only that extinction took place, not how that extinction occurred. Box B of
Figure 5 shows the monotonic decline of megafauna, using the assumptions of the
climate change and the overkill hypotheses. Box C of Figure 5 shows similar, but discrete
monotonic decline lines for both herbivores and predators, again using the assumptions of
the climate change and overkill hypotheses.
31
Table 2: Features of the Pleistocene-Holocene transition
Pleistocene Holocene
Climate Colder, less continental, higher
relative humidity, ice sheets
Warmer, more continental, lower relative
humidity, ice sheets melted
Vegetation Mixed woodland-parkland, more
patchiness in eastern and western
forests, more trees in the center of the
country
Unbroken tree cover in the east and west,
prairie-grassland in the center of the country
Animals More genera, many large animals.
more monogastrics
Fewer and smaller animals, a larger
percentage of ruminants than in the
Pleistocene
Archaeology Paleo artifacts, stylistic homogeneity,
evidence of hunting megafauna
Archaic artifacts, stylistic heterogeneity, less
megafauna
32
An Alternative Hypothesis of Extinction
An alternative hypothesis to those discussed previously proposes that the megafaunal
extinctions at the end of the Pleistocene were the result of complex interactions involving
changes in climate, changes in herbivore food supplies, changes in predator–prey
relationships, and changes in the activities of H. sapiens. In brief, the hypothesis is that
upon entry into the New World, H. sapiens reduced predator populations such that
herbivore populations expanded, which, in turn, resulted in environmental exhaustion and
ecosystem collapse due to overgrazing. As edible plants dwindled, the megaherbivores
lost their food supply and eventually became extinct by reason of starvation.
The Argument for Second-Order Predation
In support of this general approach, studies of present-day populations of herbivores
indicate that their numbers are often kept small by the action of their dependent predators
(Petersen, 1979; Carbyn et.al., 1993; Dale et.al., 1995; Seip, 1995;.Klein, 1995) Without
predator control, a population of herbivores expands to the point at which its food
becomes scarce, a situation leading to its rapid decline. If the “boom phase” of this
population cycle is large enough, significant changes within the ecosystem occur. As a
result, vegetative support for the herbivore population dwindles, and this deprivation
leads in extreme cases to a “bust phase,” which may be so severe that extinction of the
herbivore species occurs.
In the transition from the Pleistocene to the Holocene, more carnivores than herbivores
were lost (Graham, 1998). As a result, it is possible that the populations of Pleistocene
33
herbivores experienced a temporary release from control by predators. This may have led
to dramatic increases in herbivore population size, followed by such severe effects upon
and changes in the vegetation that the ecosystem was no longer able to support many of
its herbivore species. This pattern of decline is presented in Box D of Figure 5.
The boom- and-bust cycles of herbivore populations have been mathematically
described by Pitelka (1964). May (1973), Gluckenheimer et al (1976), Schaffer (1988),
and Swart (1990). They also have been observed in field studies involving caribou
introduced into predator–free islands in the Antarctic (Leader–Williams, 1988). The role
of predators in controlling herbivore populations is well illustrated by the expansion to
pest densities of white-tailed deer in predator-free suburbs of the eastern United States. It
is also evident in field studies of reindeer, caribou, moose, feral horses, burros, and bison
(Scheffer, 1951; Petersen, 1977; Leader–Williams, 1989, Carbyn et al, 1993).
An objection to this approach might be that H. sapiens would not have hunted the
predators in question. However, predators, both human and non-human, could have been
influenced indirectly by complex changes or disruptions in the ecosystem. This has been
demonstrated in some of the interactions of humans, predators, and prey in Africa. For
example, at the time of the European colonization of Africa, rinderpest, a pathogen
functioning as a predator, was introduced into the wildebeest and buffalo populations.
The loss of these animals to rinderpest was so severe that the population of lions was
deprived of a major source of food. Prior to this time, lions and humans had established a
relationship of mutual avoidance. But now, with the severe loss of their traditional prey,
the lions turned to attacking and eating humans. This threat triggered an anti–lion
34
Fig. 5 The path of extinction held by various hypotheses
35
response among humans, and the anthropogenic death of lions, though not killed for food,
contributed importantly to the reduction of their numbers (Schaller, 1972; Sinclair, 1979).
On the other hand, the reduced populations of wildebeest and buffalo, while not attaining
levels reached in pre–rinderpest times, stabilized and remained viably large (Schaller,
1972).
Another objection might be that predators do not hunt predators. However, predators
may change their behavior after a disruption in the usual predator–prey relationship. The
events surrounding the local extirpation and subsequent reintroduction of the gray, or
timber, wolf in parts of North America illustrate this pattern. Prior to the twentieth
century, foxes, coyotes, and timber wolves coexisted as predators upon a variety of
mammalian prey. During that time, foxes and coyotes avoided wolves. Perhaps because
of this avoidance, wolves were not known to kill foxes or coyotes. After the timber wolf
was extirpated from much of continental North America, generations of foxes and
coyotes hunted without competition or interference. Over the years, as they became
habituated to being the dominant predator, they apparently lost the behavior habits that
had helped them avoid the larger and more aggressive wolf. Today, when gray wolves are
reintroduced to their former lands, it is not unusual for them to kill foxes and coyotes
(Petersen, 1995). On the other hand, elk, which also experienced the absence of gray
wolves for the same amount of time, maintained the same avoidance and protection–of–
young behaviors that they had exhibited before the wolf was driven to local extinction
(Mlot, 1998). This contrast suggests that elk have a genetic avoidance response to
36
wolves, whereas foxes and coyotes seem to have learned and then forgotten their
avoidance response.
The sensitivity of predators to changes in the predator–prey relationship and the
subsequent killing of one group of predators (in the case cited, the foxes and coyotes), by
another group of predators (the gray wolves) is an example of second-order predation
(Peterson, 1995). In the instances of wolves killing foxes and coyotes, there is no
suggestion that they were killed for food. Rather, they seem to have been killed to
eliminate competition. H. sapiens, the newly-introduced, second-order predator we are
considering in the context of Holocene extinctions, did not eat other carnivores, although
these early humans may have been aware of competition, and probably killed carnivores,
utilizing their fur for clothing and their teeth for ornaments. Soffer (1985) documents
numerous instances of fur-bearing mammal bones especially the foot bones, in
Pleistocene archaeological sites (e.g. White, R. 1993. Technological and social
dimensions of Aurignacian-age body ornaments across Europe)
Second-order predation, as explained earlier, can lead to a very rapid decline in the
preyed-upon population of predators, especially if prey have no mechanisms to resist,
because of their previous status as dominant carnivores. Therefore, when non-human
predator populations are reduced, herbivore populations increase and thus trigger the
boom-and-bust cycles that result in ecosystem collapse.
In support of the theory of ecosystem collapse resulting from the impact of a
megaherbivore population explosion, the severity of such an impact on the ecosystem of
37
African megaherbivores (elephant, rhinoceros, and giraffe) has been well documented by
a variety of investigators (vanWyk & Fairfall, 1969; Laws, 1970; Leuthold, 1977; Short,
1981; Yoaciel, 1981; Anderson & Walker, 1984; Barnes, 1985; Owen-Smith, 1992). All
have found that under population stress, megaherbivores can destroy a woodland or
savanna. Anderson and Walker (1984) report that under food stress, African elephants
have turned mixed woodland into grassland by knocking over trees to get leaves at the
treetops. Wing and Buss (1970) report that the effects of elephant damage combined with
destruction by fire have converted forest into grassland in only 50 years.
According to Smithers (1983), the difference between savanna woodland and
unwooded savanna grassland is caused by over-utilization. To this researcher, his
conclusion could apply equally to the transition from mixed woodland to prairie that took
place during the Pleistocene-Holocene transition.
Proponents of the theory that environmental stress results in climate change have
observed that when trees are removed from an area to such an extent that a forest is
converted to a grassland, one result is a general drying out of the local and regional (and
perhaps even global) climate because of a loss in transpired water (Charney et al., 1975;
Potter et al., 1975). Martin (1993) writes:
With regard to transpiration, which is vaporization at the leaf surface of water
extracted from the soil by the plant, the physiological responses of the vegetation and
its physical characteristics determine the partitioning between the sensible and latent
heat at the surface of the earth. This, in turn, affects atmospheric motion, and the
water balance. (p.133)
38
Other studies indicate that there is a considerable reduction in humidity after a forest
is clear-cut. Longman and Jenik (1987) write of the effect of deforestation in Brazil:
…that the forests influence the rainfall. At 17 sites throughout the Amazon Basin, the
proportion of river water originating by evaporation from the South Atlantic Ocean or
from the forest has been estimated, using the natural frequency of the isotopes
oxygen-18 and deuterium as tracers (Salati et al. 1979). About 75 per cent of rainfall
evaporates directly or via the trees, and provides much of the moisture for cloud
formation and rain further inland. Significantly, deforestation near the coast seems to
break the cycle, which propagates the repeated succession of rainstorms moving
rapidly westwards thus threatening the survival of otherwise untouched tropical forest
ecosystems, far inland. (p.15)
A Proposed Scenario of Pleistocene Extinctions Due to Second-Order Predation
Modeling is like a laboratory experiment in that the researcher seeks to isolate the
relevant variables from extraneous factors. Economic models do this by stating at the
beginning “all things being equal,” even though everyone knows that things are never
economically equal. By the same token, real ecosystems are seldom, if ever, in a state of
equilibrium. But to isolate the relevant variables in a model of possible causes of
Pleistocene megafaunal extinction in North America, it must be assumed that prior to the
entry of new, or second-order, predators (H. sapiens), the plants, herbivores, and original,
or first-order, predators existed in a steady state—a state of dynamic equilibrium. Starting
from this steady state, the proposed scenario proceeds in the steps briefly described
below.
39
Step 1: H. sapiens Enters the New World
It is assumed that with the arrival of a new predator, namely H. sapiens, the populations
of the original, or first-order North American predators declined. This assumption is
represented graphically in Figure 6 and is based on three prior assumptions. First,
because the non-human predators did not evolve in the presence of H. sapiens, they were
naive and lacked both avoidance and defense behaviors. Second, because humans were
hunters, they frequently occupied the same territory as the non-human predators. This
propinquity gave humans the opportunity to recognize the non-human predators as actual
or potential competitors for prey species. Third, because of the pressure of an additional
predator, the prey populations declined, at least initially, thereby reducing the food
available for both humans and the non-human predators. This food shortage probably
encouraged the non-human predators to switch prey, perhaps even causing them to begin
to prey on H. sapiens.
Step 2: Second-Order Predation Begins
Because human hunters had the opportunity to learn the habits of the non-human
predators, and because they possibly saw themselves as competing with or preyed upon
by these animals, it is reasonable to assume that they eventually began to kill them. Also
since humans in cold climates require some kind of clothing, it is likely that they hunted
carnivores for their fur. At first, killing the competition relieved predatory pressure on the
prey, thus allowing the herbivores to regain at least some of their numbers. The increased
40
Fig. 6. Effect of the arrival of a new predator on the populations of North American prey and
predators
41
abundance of prey would have convinced H. sapiens that the non-human predators were
competitors that kept the food supply low. Therefore, H. sapiens would have continued to
kill them, especially if they had themselves begun to kill H. sapiens.
Step 3: Prey Populations Are No Longer Well Controlled by Predation
The continual killing non-human predators by H. sapiens is assumed to have reduced
their numbers to a point where these animals no longer regulated the size of the prey
populations.
Step 4: Prey Populations Trigger Boom-and-Bust Cycles
It is assumed that as prey populations suddenly expanded, they overgrazed and over-
browsed the land. Soon the environment was no longer able to support them. As a result,
many herbivores starved, and some species became extinct.
Step 5: Mixed Parkland Becomes Grassland
Under conditions of scarcity, mixed feeders, including mammoths and mastodons, ate
less grass and more browse (tender twigs and leaves of trees and shrubs). They pushed
over trees to get at the leaves on the tops of trees. This killed more trees. This shift in
vegetation put competitive pressure on animals that were exclusively browsers. Together,
the mixed feeders and browsers, by virtue of their eating habits, had a profound impact
on trees, eventually turning mixed parkland into prairie, or grassland.
The loss of tree cover favored the grazers, by selecting out those animals that needed
mixed food. In a denuded environment, herbivores that were able to survive on the
42
newest shoots of grass got the most nutrition out of poor quality forage. Also within
species smaller animals who could reproduce on the least amount of forage where
selectively favored. Others, such as horses, mammoths, mastodons, and sloths, could not
compete and eventually became extinct. The only surviving large herbivore was the
bison, which was smaller than ever before. It became part of a stable Holocene prairie.
The “bust” phase of the boom-and-bust cycle had placed a premium upon efficient
use of available energy, and the less-efficient monogastrics showed poorer staying power
than the more-efficient ruminants. If H. sapiens populations experienced food stress
during the time of scarcity, they would probably have strengthened their efforts to kill
any remaining non-human predators, whom they would have regarded at this point as
even more serious competitors for food. In this way, they would have unknowingly
exacerbated the problems associated with the next boom phase of the cycle.
The loss of tree cover that occurred as a result of these ecological boom-and-bust
cycles decreased atmospheric moisture, which, in turn, resulted in a more continental
climate. Continentality, as mentioned previously, is largely a factor of the relative aridity
of the environment (Deshmukh, 1986; Martin, 1993). Figures 7a and 7b show the role of
tree cover in this shift in climate. Figure 7a shows the vegetative regime under normal
conditions. Trees recruit slowly, and there is a time lag before a lost tree can replaces its
biomass. The death of an individual tree diminishes the amount of aggregate biomass of
the community of trees, but proportionally few trees die from herbivore browsing. The
obverse is true of grass. There is little loss of accumulated biomass per individual grass
plant, many grass plants suffer from herbivore grazing, and there is heavy replenishment
43
of biomass. Figure 7b shows that under scarcity conditions, there is an increase in
herbivore impact on both trees and grass, but the impact on trees is far greater because
large herbivores knock over trees to get at the tops. This kills trees for a small marginal
increase in nutrition for the herbivore.
A model can be created to test this scenario. The task of the model is to explore the
hypothesis that second-order predation resulted in an overpopulation of herbivores which
overgrazed their environment resulting in widespread extinction.
The hypothesis to be tested is: Second-order predation and its subsequent boom-and-
bust cycles explains the Pleistocene-Holocene transition better than overkill alone. The
alternative hypothesis is that overkill alone explains the data better.
44
Fig. 7. Role of trees and grass in climate change:
A. trees and grass under normal conditions; and B. trees and grass under scarcity
conditions
45
Chapter II: A Method for Testing Hypotheses
of Pleistocene Extinctions in the New World
Introduction
In the previous chapter I proposed an alternative hypothesis that might explain the
widespread megafaunal extinctions of the late Pleistocene. I suggested that extinctions
might have been brought about through second-order predation rather than simple
overkill of herbivores occasioned by the addition of late-arriving predators (i.e., H.
sapiens) to the ecosystem. Designing models that simulate both these possibilities allows
me to test the consistency of each hypothesis. One requirement for such models is that
each one use the same assumptions and starting values. Another requirement is that the
simulated environments show the impact of each possibility on plants, herbivores, and
non-human predators. An approach that fulfills these several criteria is one known as
systems dynamics.
The systems-dynamics approach addresses problems instead of solutions. Both the
Mosimann and Martin (1975) and the Whittington and Dyke (1984) models simulate a
solution to the extinction problem based solely on the overkill hypothesis. The task of
this effort is to create models of the problem that will test both the overkill and the
second-order predation hypotheses.
46
The Modeling Process
The modeling process will involve four steps and two models. The first step will be the
creation of a model that represents a simple ecosystem consisting of plants, herbivores,
and, carnivores. Because these three elements will appear together in dynamic
equilibrium, I call this stage of the model “the dynamic-equilibrium mode.” The second
step will be to elaborate the model by adding a second predator, namely H. sapiens, to the
picture. The activity at this point will be to simulate an overkill situation, in which the
second predator, in addition to the traditional predator, hunts herbivores. I refer to this
stage of the model as “the second-predator overkill mode.” Next, I will simulate second-
order predation by having H. sapiens reduce carnivore populations, a stage I call “the
second-order predation mode.” The third step will be to create a new, more complex
model, by disaggregating plants into trees and grass, and herbivores into browsers,
grazers, and mixed feeders. I will run this model as well in dynamic-equilibrium, second-
predator overkill, and second-order predation modes. The fourth step will be to elaborate
on this second model by dividing grazers into ruminants and non-ruminants. I will run
this elaborated model in the three modes described above.
I will introduce these models according to the following procedure. First, I will
present general conventions and definitions that apply to the entire modeling effort. Next
will come an overview of the process. Following the overview, I will give a step-by-step
explanation of the simulation. At the beginning of each step, I will provide the relevant
procedures and conventions and then present the relevant equations. At the end of each
step, I will describe results.
47
General Conventions and Definition of Terms
The following conventions and definitions apply throughout the modeling effort:
The term model, as used here, signifies a simplified representation of some aspect of
reality.
The term feedback is defined as the transmission and return of information to the
model. Specifically, the model generates information, which it can use at a later time. For
example, consider a hypothetical model that shows a specific number of organisms in
existence in year one. During the course of that year, some of the organisms reproduce
and some die. Thus, in year two, there will be the original number of organisms, plus the
number who were born, less the number who have died. This new number, from which
we can now determine the population growth rate, is the result of feedback. The
population events that took place during the first year have been “fed back” into the
model for determining the size of the population in the second year.
Reference mode is the way in which a graph reflects either observed or inferred
behavior of certain key variables. If the behaviors are observable, then the graph is said to
have an observed reference mode. If the behaviors cannot be observed, then the graph is
said to have an inferred reference mode. In other words, it reflects a hypothesis about the
behaviors of certain variables over time. The reference modes for this modeling effort are
presented below.
The inferred reference mode for the model before the introduction of H. sapiens is
one of dynamic stability. Graphically, this suggests that the lines representing the size of
48
all populations over time should be flat. However, each population should respond to a
disturbance in one of the populations, as shown in Figure 8.
The inferred reference mode for second-predator overkill is shown in Figure 9.
The inferred reference mode for the second-order predation is shown in Figure 10.
Elaboration is a term meaning an addition or amendment to a previous model. It
implies a contrast with the original design. Step 1 in this modeling process is an original
design. Step 2 is an elaboration of Step 1, by virtue of adding H. sapiens to the model.
Step 3, on the other hand, is a new model, which was designed using the values and
thinking from Steps 1 and 2. Step 4 is an elaboration of Step 3.
The term stock refers to the net production of the unit in question. Thus, “herbivore
stock” signifies the number of herbivores at any given time.
Flow is defined as a change in the size of a stock. For example an increase to the
stock of herbivores is called an inflow; a decrease is called an outflow.
Time (t) is the term used to mean the time currently under consideration. In this model,
time is measured in years before present.
Delta time (dt) is the increment of time used. In this model, the increment of time
used is one half of one year.
Sector is the term used for all the equations and values that pertain to a certain stock
or stocks. For example, the sector for a stock of small and big trees would include their
associated recruitment rates and death rates, along with the equations apportioned to their
49
Fig. 8. Dynamic equilibrium reference mode
50
Fig. 9. Second-predator (overkill) reference mode
51
Fig. 10. Second-order predation reference mode
Key as in figure 9
52
respective carrying capacities. In diagrams, I use broken lines as a convenient way to
indicate sectors.
Graphical function is used in this modeling language to specify a relationship
between two variables on an x–y coordinate system. It is used for relationships that may
be non-linear. For example, if one believes that hunting by carnivores decreases when
there are more herbivores readily available, then one will describe the hunting effort as
tapering off when the density of herbivores is near or at its maximum.
Step addition or step reduction is defined as a one-time increase or decrease of a
specific amount that occurs at a specific time. Both possibilities are shown graphically in
Figure 11.
Pulse addition or pulse reduction is defined as a temporary increase or decrease of
some specific amount for a time period that lasts for a specific interval. Both situations
are shown graphically in Figure 12, along with alternative paths of displacement and
recovery.
Delay or lag describes a delay of a specified unit of time or delta time. For example,
the hunting rate at any given time is based on the density of herbivores at two delta-time
periods in the past (t–2dt). Thus, if the number of herbivores has decreased in the last unit
of time (t–dt), the hunting pressure is delayed until the next time period.
A switch refers to conditions under which an equation may change. This switch takes
place when an equation contains an “if” statement. “If” statements activate one condition
or another, depending on the response to the “if.” For example, for variable Y, if X equals
53
Fig. 11. Illustration of the step function
54
Fig. 12. Illustration of the pulse function
55
0, do A; if X does not equal 0, do B. The switch indicates the conditions under which one
does one or the other, A or B. In this work, a switch is indicated in plain (Roman)
typeface. The ordinary operation of the equation is indicated in italics. This format for the
equation of the example given above is Y = If X = 0 then A else B
The abbreviation a.u. stands for “animal unit.” An animal unit is equal to 1,000
pounds (approximately 450 kilograms) of herbivore, based on calculations formulated by
Whittington and Dyke (1984), as cited earlier.
Initial values is the term used for the starting values of stocks. The values of all
stocks are initialized at equilibrium using the following procedure. Plants are initialized
at 25 a.u. of plants for each unit of area, as per Whittington and Dyke (1984). Then the
model is run starting with 10 a.u. of herbivore and 1 a.u. of carnivore until all the stocks
reach equilibrium. These values are used as initial values in all subsequent runs of the
model.
The term area refers to the amount of land arbitrarily assigned to the model’s
continent, namely 30,000 square miles.
Values used in these simulations are values used in the simulations that are based
directly on those used by Whittington and Dyke (1984). They are shown in Table 3
below.
56
Table 3: Values taken directly from Whittington and Dyke (1984)
DESCRIPTION OF
VARIABLE
BASELINE SOURCE
Human population size 200 Budyko 1967, 1974
Human population growth rate 0.0443 Birdsell 1957
Table 4 shows starting values that were modified to fit the modeling paradigm.
Table 4: Modified values based on Whittington and Dyke (1984)
DESCRIPTION OF
VARIABLE
BASELINE VARIATION
Prey carrying capacity (a.u.*
per square mile)
25 Changed to the amount of (plants) per square
mile.
Prey biomass replacement
(a.u.* per square mile)
0.25 Used as base recruitment rate: birth rate minus
average death through hunting by predators
plus natural death rate = 0.25
* animal units = 1K lb. of herbivore
K : Whittington and Dyke (1989) assign 25 herbivore units to 1 unit of area (see
above). This suggests that there are 25 units of plants per unit of area. This is the
maximum carrying capacity of the continent at equilibrium.
Carnivore recruitment is the net growth of carnivore population (births minus
deaths) in any given year. From informal estimates based on extant predators (Carbyn,
1994; The Cat House, 1997; International Wolf Center, 1997), carnivore recruitment is
10 percent a year.
57
The hunting rate used in the simulation is based on food needed per pound of
predator per year. Data from extant predators suggests 20 pounds of food per year per
pound of predator (Schaller, 1972; Petersen, 1977; International Wolf Center, 1996; The
Cat House, 1996). The maximum number of herbivores per unit of area is set at 25
a.u.(MaxHrbP = 25), following Whittington and Dyke (1984)
General Overview of the Model
The task of the model is to explore the hypothesis that second-order predation resulted in
an overpopulation of herbivores, who overgrazed their environment. Overgrazing
eventually resulted in widespread extinction. Stated another way:
Second-order predation and its subsequent boom-and-bust ecological dynamic
explain the data better than overkill alone? The alternative hypothesis is that overkill
alone explains the data better.
To test the second-order predation hypothesis the models created must be able to
simulate the conditions of both overkill and second-order predation, using the same
starting assumptions and values.
I produced two models in a four–step process. The first two steps involved
constructing a model that is an aggregated simulation of a simple ecosystem consisting of
five components: a continent, plants, herbivores, predators, and H. sapiens. The second
model is a more complex, disaggregated simulation, in which I performed the third step
by disaggregating the plant component of the first model into trees and grass, and
disaggregating the herbivore component into browsers, grazers, and mixed feeders. The
58
fourth step was the same as the third, except that I disaggregated the grazers into
ruminants and monogastrics.
The Simple Model
The purpose of the first step of the modeling activity is to establish the basic model. The
purpose of the second step is to show the dynamics of the introduction of a second
predator and to contrast that with the dynamics of second-order predation.
The Disaggregated Model
The third step examines how predatory/prey dynamics would affect the vegetation and
herbivores in a more complex system. The fourth step examines how changes proposed in
the previous steps affect ruminant and monogastric grazers.
Criteria for Success
I will consider the modeling and simulation project a success if it indicates the conditions
under which the second-order predation hypothesis would have operated. If these
conditions are found to be less likely to produce extinction than those required for the
overkill hypothesis then the second-order predation hypothesis will be rejected.
59
Base Model: Dynamic Equilibrium – Step 1
Overview
I will start with an initial model based on the Lotka–Volterra predator-prey equations
(May, 1976; Schaffer, 1988; Smuts, 1979; Hilborn & Sinclair, 1979; Richardson and
Pugh, 1981).
Model Diagram
The model diagram below (Figure 13) illustrates the relationship between the various
sectors. Each stock is limited by the stock above it, which serves as its carrying capacity.
Thus Continent is the carrying capacity for Plants, Plants are the carrying capacity for
Herbivores and Herbivores are the carrying capacity for Carnivores. The box labeled
density shown at the bottom of the diagram contains copies of the stocks, and auxiliary
values and equations, which control herbivore hunting by carnivores. It is separate for
ease in understanding the diagram and does not constitute a separate trophic level.
Each limit is used up by the stock, which it limits. Thus Plants fill up the available
area, Herbivores eat plants, and Carnivores kill Herbivores.
This is the base model upon which other elaborations and adjustments will be made.
It should behave such that change in either the carnivore or the herbivore populations
results in a parallel change in the other.
60
Fig. 13. Base model diagram.
61
Fig. 13. continued
62
Conventions, Definitions and Equations
Continent
Area = Size of continent = 30,000mi2
HerbPerArea = Density of herbivores that can be supported by plants = 25 animal units
per square mile (where 1 animal unit equals the amount of plants required to
support 1,000 pounds of herbivore for one year)
AreaMultipleHrb = Carrying capacity of the continent for herbivores = Area *
HerbPerArea = 750,000 animal units of herbivore.
Plants
Plants is the stock of plants which is equivalent to the amount of plants at the previous
time plus the input to plants minus the outflow of plants times the unit time;
Plants(t) = Plants (t–dt) + (InPlants – OutPlants) * dt
AreaMultipleHrb is equivalent to the density of plants, which is equivalent to the number
of plants per unit (expressed as a number of animal units of herbivore that can be
supported by 1 square mile) (by assignment based upon Whittington and Dyke
(1984))
HerbPerArea =25 Area * HerbPerArea is equivalent to the maximum stock of plants
AreaMultipleHrb = 30,000 * 25=750,000
63
PlantRegen is equivalent to the replacement rate of plants PlantRegen = 1 (by
assignment)
(1–PlantRegen/AreaMultipleHrb) is equivalent to the maximum fraction of plants that
can be added to the continental stock of plants (expressed in number of animal
units). It is equivalent to the limit to plant growth which is equivalent to the
remaining fraction of continental carrying capacity that can be filled (i.e. that is
empty)
InPlants is equivalent to the number of plants added to plant stock in time interval
(=1year)
Plants * PlantRegen * (1–Plants/AreaMultipleHrb)
PlantEating is equivalent to the amount of plant material lost from plant stock (Plants).
PlantEating is in support of the population of herbivores (1 unit of plant eating =
the amount of plants required to support 1 animal unit of Herbivore when
measured in animal units = number of herbivores (which is equivalent to
Herbivore) in animal units.
OutPlants is equivalent to the amount of plant material (in animal units) removed from
Plants. Since plants replace themselves (PlantRegen=1), the only loss of plant
material is PlantEating and so OutPlants = PlantEating
64
Herbivores
Herbivores – the stock of herbivores which is equivalent to the number of herbivores
present at any given time, expressed in animal units;
Herbivores (t)=Herbivores (t–dt) + (InHrb–OutHrb) * dt
BrateH – the birth rate of Herbivores; reflects the disaggregation of recruitment used by
Whittington and Dyke (1984) into birth rate, death rate, and death by hunting. Set
arbitrarily at 0.9 individuals per individual in the population of herbivores.
(1–(Herbivores/Plants)) is equivalent to the limit to herbivore population growth imposed
by the carrying capacity of the continent for plants; which is equivalent to the
maximum fraction of herbivores that can be added to the continental stock of
herbivores; expressed in animal units.
InHrb is equivalent to the number of herbivores added to herbivore stock per unit time;
=Herbivores * BrateH * (1–(Herbivores/Plants))
DrateHrb is equivalent to the non- hunting death rate of Herbivores = 0.12 (arbitrarily
assigned)
MaxHerbA is equivalent to the maximum number of animal units a unit of area can
support = 25 (arbitrarily assigned by Whittington and Dyke (1984)).
HerbDensity is equivalent to the actual density of herbivores per unit area:
Herbivores/Area
65
DeltaHerbPerArea is equivalent to the difference between the actual density of
Herbivores (HerbDensity) and the maximum herbivore density (MaxHerbA):
HerbDensity/MaxHerbA
HuntingGrf is the rate at which predators kill herbivores. The input to the graph
(DeltaHerbPerArea) (shown on the X-axis) determines what value of HuntingGrf
(shown on the Y-axis) will be returned as output to the model. As herbivore
density approaches the maximum herbivore density carnivores are able to kill
more herbivores. Because it is assumed at maximum herbivore densities carnivore
hunting will be very successful and by the same token at low densities carnivores
are required to hunt hard. The graph of this function is shown in Figure 14
FoodNeedCrn is equivalent to the amount of food (animal units) needed to support a
pound of carnivore per year = 20 lbs. (Carbyn, 1995; The Cat House, 1997;
International Wolf Center, 1997).
HuntingC is equivalent to the loss of herbivore due to consumption by a single carnivore;
FoodNeedCrn * HuntingGrf * Herbivores
HerbKilledC is equivalent to the loss of herbivores due to hunting;
HuntingC * Carnivores
OutHrb is equivalent to the loss of herbivores from the stock of herbivores per unit time;
DrateH + HerbKilledC
66
Fig. 14. Hunting function
HuntingGrfCrn = GRAPH(DeltaHCrn)
(0, 0.000); (0.1, 0.150); (0.2, 0.280); (0.3, 0.390); (0.4, 0.490); (0.5, 0.590); (0.6, 0.675); (0.7,
0.760); (0.8, 0.846); (0.9, 0.925); (1, 1.000)
67
Carnivores
Carnivores is equivalent to the stock of carnivores;
Carnivores (t) = Carnivores (t – dt) + (InCrn – OutCrn) * dt
BrateCrn is equivalent to the natural birth rate of carnivores arbitrarily assigned at 0.4
individuals per individual in the population of carnivores = 0.4 (arbitrarily
assigned)
HerbDelay is equivalent to the amount of time when the population of Carnivores
responds to a change in the Herbivore population. If HerbDelay is 2 the impact of
a change in the population level of the Herbivore stock does not impact
Carnivores until 2 time periods after it occurs; time (t) minus two delta-time(2dt).
HerbDelay = 1
CarnivoreK is equivalent to Herbivores at the time period specified (t–2dt) by HerbDelay
the amount of time lag specified in HerbDelay.
(1–(Carnivores/CarnivoreK)) is equivalent to the limit to carnivore population growth
imposed by the size of the Herbivore population equivalent to the maximum
fraction of Carnivores that can be added to the stock of carnivores: expressed in
animal units.
InCarn is equivalent to the number of carnivores added to Carnivore stock per unit time:
Carnivores * BrateCrn * (1–(Carnivores/CarnivoreK))
68
DrateCrn is equivalent to the natural death rate of carnivores = 0.3 (arbitrarily assigned)
OutCrn is equivalent to the loss of carnivores from the stock of carnivores.
The model will be considered to be in dynamic equilibrium if a disturbance to any of
the stocks returns to the model to equilibrium. For the model to be judged sufficiently
linked, a pulse disturbance of any of the stocks should have an impact on the stocks with
which it is linked. This is necessary in order to establish a baseline of steady state so that
we know that the disturbances which follow are due to our experiments not to some
aberration in the model it self.
Graph of the Base Model – Step 1
Figure 15 – Graph of the Base Model – shows that when the model is run it is in
equilibrium (Sensitivity tests of step 1 are found in Appendix A).
In the graphs presented below, the model is perturbed by creating a pulse outflow of
5% of the population of the stock starting at –11,500 and again 250 years later, it rapidly
returns to stability. Figure 16a –5% Pulse outflow of Plants, 16b –5% Pulse outflow of
Herbivores, and 16c – 5% Pulse outflow of Carnivores, show the effects of perturbing
each of the stocks in turn.
Note when the model is disturbed higher on the food chain it causes more instability
and it takes longer for the model to return to stability than when it is disturbed lower on
the food chain. Disturbing the Carnivore stock by causing a pulse outflow of 2.5% creates
69
Fig. 15. Graph of the base model
Key as in figure 9
70
Fig. 16. Pulse outflow from plants, herbivores and carnivores
Key as in figure 9
A.
B.
71
Fig. 16. continued
C
72
more instability for a longer period of time than disturbing the Herbivore and more again
than disturbing Plants.
Second Predator: Overkill – Step 2a
Overview
To test the consistency of either of the hypotheses (overkill and second-order overkill)
with extinction the base model was elaborated.
This step, the first elaboration, introduces a second predator, H. sapiens. This is the
position of the overkill hypothesis suggested by Martin (1967, 1984), Mosimann and
Martin (1975), and Whittington and Dyke (1984).
Model Diagram for Step 2a
The illustration shown below (Figure 17 – Model Diagram) is the Base model from Step
1 with the addition of a sector for H. sapiens. The stock of H. sapiens (Hsapiens) inflow,
outflow and hunting equations have the same form as the corresponding equations for
Carnivores from the Base model explained in Step 1. The major difference between the
sectors is that Hsapiens is initially an empty stock. Therefore, it is necessary to create
equations, which simulate the migration of H. sapiens into an ecosystem in equilibrium.
The density box that modifies the impact of hunting on Herbivores has been changed
to add the appropriate equations and values to account for hunting by H. sapiens.
73
Fig. 17. Second predator (overkill) model diagrams
Key as in figure 13
74
Conventions, Definitions, and Equations
Modifications to Density and Herbivore Sectors
FoodNeedHs is equivalent to the amount of food (animal units) needed to support a
pound of Hsapiens per year = 10 lbs. It is arbitrarily assigned as half the food
needed by a Carnivore.
HuntingHs is equivalent to the loss of herbivore due to consumption by a single
Hsapiens;
HuntingHs = FoodNeedHs * HuntingGrf * Herbivores
HerbKilledHs is equivalent to the loss of herbivores due to hunting by Hsapiens;
HerbKilledHs = HuntingHs * Hsapiens
HerbsKilled is equivalent to the sum of the loss of herbivores due to hunting by Hsapiens
and by Carnivores;
HerbsKilled = HerbKilledC + HerbKilledHs
OutHrb from step 1 has been modified by using HerbKilled instead of HerbKilledC it is
equivalent to the loss of herbivores from the stock of herbivores per unit time;
OutHrb = DrateH + HerbKilled
75
Hsapiens
Hsapiens is equivalent to the stock of H. sapiens. Since the hypothesis we are testing is
the inmigration of H. sapiens it is initialized at zero (0)
Hsapiens (t) = Hsapiens (t – dt) + (HsMigrate + InHs – OutHs) * dt
AmtHsMigrate is equivalent to the number of H. sapiens entering the continent.
Whittington and Dyke (1984) arbitrarily assign it at 200 individuals.
BiomassHs is the conversion of number of H. sapiens into biomass. It is assigned a value
of one hundred pounds (100 lbs.)
TimeMigrateHs is equivalent to the time at which H. sapiens enters the continent. It is
arbitrarily assigned to –11500 BP
HsMigrate is equivalent to the amount the stock is filled using with a one–time inflow
(pulse) of H. sapiens in biomass (BiomassHs) at –11500 years ago
(TimeMigrate). Because there is no repeat of the pulse there is a zero (0)
following the TimeMigrateHs:
HsMigrate = PULSE (BiomassHs, TimeMigrateHs, 0)
BrateHs is equivalent to the birth rate of H. sapiens; it reflects the disaggregation of
recruitment used by Whittington and Dyke (1984) into birth rate, death rate, and
death by hunting. Set arbitrarily at 0.047 individuals per individual in the
population of Hsapiens.
76
(1–(Hsapiens/Herbivores)) is equivalent to limit to H. sapiens population growth
imposed by the size of the Herbivore population equivalent to the maximum
fraction of Hsapiens that can be added to the stock of H. sapiens: expressed in
biomass.
InHs is equivalent to the number of H. sapiens added to H. sapiens stock per unit time.
Because the Hsapiens stock is initialized at zero (0) it is necessary to put a switch
in the equation which allows the model to function with a zero stock. It is shown
in plain text:
If Hsapiens = 0 then 0 else Hsapiens * (BrateHs * (1–Hsapiens/Herbivores))
DrateHs is equivalent to the natural death rate of H. sapiens = 0.03 (arbitrarily assigned)
OutHs is equivalent to the loss of H. sapiens from the stock of H. sapiens.
Results Second Predator (Overkill) – Step 2a
This elaboration will be a success if we see the impact of the introduction of a second
predator This is the position of the overkill hypothesis suggested by Martin (1967, 1984),
and Mosimann and Martin (1975), and Whittington and Dyke (1984).
Graph of the Model – Step 2a
In the graph of the model (Figure 18 – Graph of the Second Predator (Overkill) Model)
below we see the impact of the introduction of a second predator into the ecological
system. Herbivores and carnivores decreased and plant stock increased.
77
Fig. 18. Graph of the second predator (overkill) mode
Key as in figure 9.
78
Second-Order Predation – Step 2b
Overview
This elaboration of the initial model is to simulate second-order predation. The values
and the components of the second predator model are the same the difference between the
two models is the addition of predator/predator hunting values.
Model Diagram
The diagram, (Figure 19) shows the new outflow from Carnivores.
Conventions, Definitions, and Equations
All conventions and definitions from Step 1, 2a and the general conventions and
definitions section remain the same.
AmtKill is the percent of the reduction of Carnivore populations by Hsapiens. The default
value is 1.5%. It was arrived through trial and error as a small percentage
reduction, which still had a significant impact.
KillTime is the time after migration that H. sapiens (Hsapiens) begins to reduce
Carnivore populations. The default is arbitrarily assigned to 300 years.
OutStepCrn is equivalent to the amount Carnivore populations are reduced by Hsapiens,
at an appointed time after the inmigration of Hsapiens. OutStepCrn has a switch
(shown in plain text) which allows the model to function when there are no
79
Fig. 19. Second-order predation diagram
Key as in figure 13
80
Hsapiens present. The equation function is shown in italic text:
OutStepCrn = If Hsapiens=0 then 0 else STEP (AmtKill, (TimeMigrationHs,
KillTime))
Carnivores is the stock of carnivores. It has been modified to include the additional
outflow so that the equation now reads:
Carnivores (t) = Carnivores (t – dt) + (InCrn – OutCrn – OutStepCrn) * dt
Results Second-Order Predation – Step 2b
Graph of Model – Step 2b
The graph below (Figure 20 – Second-Order Predation) shows the impact of a small
(1.5%) reduction in Carnivore populations on the ecosystem as a whole. There is an
immediate population explosion of herbivores and an immediate reduction in the amount
of plants. The slope of the reduction of Carnivores is affected very little. The
disequilibrium in the plant and in herbivore populations is the most salient feature of the
reduction of Carnivore populations.
Table 3 shows there are slightly fewer herbivores in the second predator model (Step
2a) than in the second-order predation model (Step 2b). H. sapiens does better in the
second-order model with an ending value of 15,194.47 as opposed to 13,922.97 in the
second predator model or approximately 1 person per every 2 miles.
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Fig. 20. Graph of the second order predation model
Key as in figure 9
82
Table – 3 – Comparison of Second Predator (Overkill) and Second-OrderPredation Ending Values
Starting Value Second Predator Second-Order Predation
Plants 638,439.63 703,337.67 698,741.58
% starting value 110% 109%
Herbivores 94,966.08 43,759.15 47,755.17
% starting value 46% 50%
Carnivores 16,619.06 7,657.91 6,566.40
% starting value 46% 40%
Hsapiens 13,922.97 15,194.47
83
Step 3 – Three Herbivores – (Browsers, Grazers and Mixed Feeders)
Overview
Step three models a more complex ecosystem where the two hypotheses (overkill and
second-order overkill) can be tested. Since the step two model does not address the
pattern of extinction it was necessary to make a more complex model. Trees and Grass
have replaced plants. This made changes in the Continent sector necessary. Browsers,
grazers and mixed feeders, have replaced herbivores. Hunting equations have been
changed to reflect the division of herbivores and to introduce prey switching. The hunting
rate graphs have also been made more realistic. It is assumed that because carnivores are
obligate predators the hunting rate of carnivores will be more closely tied to prey density
whereas H. sapiens is an omnivore so it is assumed that at low prey density H. sapiens
hunting will drop off. Therefore the graph of Hsapiens hunting rates is more sigmoid
whereas the equivalent graph for Carnivores is linear. In the step 2 model carnivore
populations were reduced by a percentage each year. In this model the reduction in
carnivore populations is a factor of carnivore density and a kills per Homo sapiens. All
conventions and definitions, not discussed, remain as previously specified.
Diagram of the Model
An illustration of the elaboration of the continent, tree and grass equations is shown in
Figure 21
84
Fig. 21. Continent, trees and grass, diagram. Three herbivore model.
Key as in Base Model diagram.
85
In the base model hunting was a factor of the difference between the actual density of
herbivores and the maximum number of herbivores. The maximum number of herbivores
is the amount necessary for the predator (human or non-human) to achieve maximum
reproduction. In this model hunting is based on actual density of herbivores. An
illustration of the relationships specified by the density equations is shown in Figure 22.
An illustration of the redesign of Herbivores for step three is shown in Figure 23.
In the base model the reduction in carnivore populations was a percentage of the
population per year. It was designed to illustrate how little carnivore populations needed
to be reduced in order to change the balance in the ecosystem. In this model the reduction
in carnivore populations is due to the density of carnivores and H. sapiens hunting
carnivores at a set rate. An illustration of the redesign of Carnivores and Hsapiens is
shown in Figure 24.
Conventions, Definitions, and Equations
Continent
AllTrees is the sum of small and big trees;
AllTrees = BigTrees + SmallTrees
Plants is the sum of all the Plants; small and big trees, high and low quality grass;
Plants = BigTrees + SmallTrees + GrassHigh + GrassLow
86
Fig. 22. Herbivores. – browsers, grazers, and mixed feeders diagram. Three herbivore model.
Key as in Base Model diagram.
87
Fig. 23. Density diagram. Three herbivore model.
Key as in Base Model diagram.
88
Fig. 24. Carnivores, Hsapiens diagram. Three herbivore model.
Key as in Base Model diagram
89
TreePerC is the ratio of all trees to all Plants subtracted from one (1); as the number of
trees increase TreePerC decreases such that if the continent is all trees the
equations returns zero (0) and if it is all grass the equations returns one (1);
TreePrC = 1–AllTrees/Plants
WoodMix is equivalent to how much of the continent (K from the previous models) is
allocated to trees based on how much it was the previous time period. The shift in
environments is relatively slow so there is a large flat spot in the center of the
graph that preserves stability. When the extremes are reached then the system
should shift fairly quickly. Therefore, the slope of the curve at either end is more
extreme. The input to the graph is TreePerC. A graph of the function is shown
below (Figure 25);
WoodMix. = GRAPH (TreePrC)
TreeK is the portion of the carrying capacity that is allocated to trees by the graphical
function above:
TreeK = K * WoodMix
GrassK is the portion of the continent that is allocated to grass by subtracting the amount
of the carrying capacity allocated to trees from 1:
GrassK=1–TreeK
Plants
There are two sectors to Plants Trees and Grass.
90
Fig. 25. WoodMix function. Three herbivore model.
WoodMix=GRAPH(TreePrC)
(0.4, 0.697); (0.46, 0.438); (0.52, 0.354); (0.58, 0.319); (0.64, 0.301); (0.7, 0.270); (0.76, 0.249);
(0.82, 0.231); (0.88, 0.196); (0.94, 0.144); (1, 0.000)
91
Trees
Note: Small trees are the preferred food of browsers and mixed feeders. It is assumed that
small trees provide more energy per unit of biomass than big trees. If the demand for
trees exceeds the stock of small trees, big trees are eaten (see BtEating below). Eating big
trees reduces the biomass of big trees more than eating small trees reduces the biomass of
small trees. Each unit of nutritional demand for big trees has a 1.5 unit impact (see
below). This is based on the observation by Wing and Buss, (1970) of elephants under
population stress killing mature trees by knocking them over to eat the leaves at the top.
Trees. The tree sector has two stocks, small trees (SmallTrees) and big trees (BigTrees).
SmallTrees recruit quickly. The SmallTrees stock includes new shoots to existing
trees, some of them die through being eaten, natural death, and some mature into
BigTrees and ultimately die. All tree reproduction comes through small trees.
Both small and big trees contribute to reproduction. Trees are eaten by herbivores,
small trees are preferred to big trees.
SmallTrees is equivalent to the stock of SmallTrees; the amount of SmallTrees present at
any given time that can be supported at that time;
SmallTrees(t) = SmallTrees (t–dt) + (InTrees – Maturity – OutSt) * dt
Reseed is used in a switch to replant trees if all tree stocks are depleted. It is only active
under extreme conditions. It introduces 1 unit of trees for use in reproduction.
92
StRepo is the proportion of reproduction attributed to small trees. It is arbitrarily assigned
a rate of 0.6.
BtRepo is the proportion of reproduction attributed to big trees. It is arbitrarily assigned a
rate of 0.4.
TreeRepo is the sum of reproduction attributed to small trees and big trees:
TreeRepo= StRepo + BtRepo
TreeRepRate is the reproduction rate of trees set to 0.25. This is based on loosely on the
following thinking. Whittington and Dyke (1984) used a constant or fixed
carrying capacity. Their herbivore recruitment was 0.25 therefore the carrying
capacity of the environment must be sufficient to support that amount of
recruitment.
(1–AllTrees/TreeK) is equivalent to the maximum fraction of Trees that can be added to
the continental stock of Trees. It is expressed in the number of animal units that is
equivalent to the limit to Tree growth. This is equivalent to the fraction of
continental carrying capacity allotted to trees remaining, (TreeK) that can be filled
(i.e. that is empty).
InTrees is equivalent to the number of Trees added to Trees stock in time interval (=1
year); It has a switch (presented in plain text) which is activated when there are no
trees left on the continent otherwise the equation functions as it appears in italic
text.
93
InTrees = If AllTrees < 0.001 then Reseed else TreeRepo * (TreeRepRate * (1–
AllTrees/TreeK))
StDeath is the non- herbivore related death rate of small trees. It is arbitrarily assigned a
rate of 0.05.
BzEfficiency is equivalent to Browser efficiency measured in animal units
Efficiency is a measure of how much food it takes to produce one animal unit
(a.u.). Thus an “average” animal unit will have an efficiency of 1; it will require 1
unit of food. A more efficient animal will be able to produce more animal per unit
of food. Its efficiency will be less than 1. A less efficient animal will require more
than one unit of food so it’s efficiency will be greater than 1. Browser efficiency
is arbitrarily assigned a value of one; BzEfficiency=1.
TreesNeedBz is equivalent to the number of browsers multiplied by the efficiency of
browsers which is 1 so the trees needed is equal to the number of Browsers
TreesNeedBz = Browsers * BzEfficiency
MxTreePerC is equivalent to the percentage of trees in the diet of MixedFeeders = 0.5
(arbitrarily assigned).
TreeRatio has a switch in it to set the ratio to zero when there are less than 10 trees. This
becomes active only under extreme conditions TreeRatio is equivalent to the ratio
of trees to AllTrees plus AllGrass. = AllTrees/(AllTrees+AllGrass).
94
MxEfficiency is equivalent to mixed feeder efficiency. The optimum efficiency for
MixedFeeders is when the number of trees in the environment is the same, as the
number needed by MixedFeeders. Deviation from this point decreases
MixedFeeders efficiency. Thus a graph of this function is a ‘U’ shaped graph with
the optimum efficiency at the base of the ‘U’. Deviation from the optimum on
either side of the ‘U’ increases the amount of food necessary to obtain the same
nutritional benefit. The graph takes TreeRatio as its input and puts out the
efficiency of MixedFeeders. The graph of the function is shown in Figure 26
MxEfficiency = GRAPH (TreeRatio)
TreesNeedMx is equivalent to the amount of trees eaten by MixedFeeders at any given
time
TreesNeedMx = MixedFeeders * (MxEfficiency * MxTreePerC)
BzMxTreeNeed is the sum of trees needed by Browsers and MixedFeeders;
BzMxTreeNeed = TreesNeedBz + TreesNeedMx
StEating has a switch (shown in plain text) that becomes active when the amount of small
trees needed (BzMxTreeNeed) is greater then the stock of SmallTrees then
SmallTrees is used up otherwise StEating is equivalent to the amount of trees
needed (BzMxTreeNeed).
StEating = If BzMxTreeNeed>SmallTrees then SmallTrees else BzMxTreeNeed
95
Fig. 26. MixedFeeder efficiency. Three herbivore model.
MxEfficiency = GRAPH(TreeRatio)
(0, 4.00); (0.1, 2.75); (0.2, 1.60); (0.3, 1.25); (0.4, 1.05); (0.5, 1.00); (0.6, 1.05); (0.7,
1.25); (0.8, 1.60); (0.9, 2.75); (1, 4.00)
96
OutSt is equivalent to the outflow from small trees which is lost to the tree sector (i.e. it
does not pass into big trees);
OutSt = (SmallTrees * StDeath) + StEating
MatRate is equivalent to the rate at which small trees mature into big trees; it is arbitrarily
assigned a rate of 0.15.
Maturity is the number of small trees become large trees;
SmallTrees * MatRate
BigTrees is equivalent to the stock of BigTrees; the amount of BigTrees present at any
given time;
BigTrees (t) = BigTrees (t – dt) + (Maturity – OutBt) * dt
BtRatio is equivalent to the amount of increased impact on big trees when Browsers and
MixedFeeders are forced to eat a less preferred food, BigTrees. This reflects the
need for herbivores to kill more biomass of big trees to obtain the same
nourishment. It is equal to 1.5 (arbitrarily assigned)
AmtBt multiplies the amount of tree demand not filled by small trees by the amount of
increased impact (BtRatio). It is determined by subtracting the amount of
SmallTrees from the amount of tree demand (BzMxTreeNeed) and multiplying the
remainder by the amount of increased impact from BtRatio
AmtEatBt = (BzMxTreeNeed –SmallTrees) * BtRatio
97
BtNeed evaluates if there are more trees needed by all herbivores (BzMxTreeNeed) than
the stock of SmallTrees. If there are it calls AmEattBt since this is not the case
under equilibrium conditions the default is zero (0).
BtNeed = If BzMxTreeNeed > SmallTrees then AmtEatBt else 0
BTEating evaluates if the amount of BigTrees needed (BtNeed) is greater then BigTrees
if so it uses the entire stock of BigTrees if not it calls BtNeed.
BtEating = If BtNeed > BigTrees then BigTrees else BtNeed
BtDeath is the non- herbivore related death rate of big trees. It is arbitrarily assigned a
rate of 0.03.
Grass
Grass has two levels: high quality grass (GrassHigh) and low quality grass (GrassLow).
Recruitment is into the GrassHigh level but both kinds of grass participate in
reproduction. Some GrassHigh passes into GrassLow, some is eaten and some dies a
natural death. Grazers and mixed feeders prefer high quality grass to low quality grass. If
the demand for high quality grass exceeds the stock of high quality grass, low quality
grass is eaten. More low quality grass is necessary to provide the same level of
nourishment as high quality grass.
AllGrass is equivalent to the sum of high and low quality grass
AllGrass = GrassHigh + GrassLow
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High Quality Grass
GrassHigh is equivalent to the stock of high quality grass which is equivalent to the
amount of high quality grass that is present at any given time;
GrassHigh (t) = GrassHigh (t – dt) + (InHi – OutHi – HiToLo) * dt
HiRepo is equivalent to the amount that GrassHigh contributes to the total reproduction
of grass. It is considered to be half the contribution = GrassHigh * .5 (arbitrarily
assigned)
LoRepo is the proportion of reproduction attributed to GrassLow. It is considered to be
half the contribution = GrassLow * .5 (arbitrarily assigned)
GrRepo is the sum of high and low quality grass to reproduction;
GrRepo= HiRepo + LoRepo
RateGr is the reproduction rate of grass it is arbitrarily assigned a rate of 0.33.
(1–AllGrass/GrassK) is equivalent to the maximum fraction of grass that can be added to
the continental stock of grass. It is expressed in number of animal units equivalent
to the limit to grass growth, which is equivalent to the fraction of continental
carrying capacity allotted to grass left, (GrassK) that can be filled (i.e. that is
empty).
InHi is equivalent to the amount of high quality grass that is added to high quality grass
stock in time interval (=1year)
GrRepo * (RateGr * (1–AllGrass/GrassK))
99
SetEffGz is equivalent to the setting for optimal Grazer efficiency it is arbitrarily assigned
a value of 0.9
FoodNeedGz is equivalent to the amount of food necessary to support the population of
grazers;
= Grazers * SetEffGz
GzGrass is the ratio of how much food is needed by grazers to the amount of grass
present in the system subtracted from one;
1–FoodNeedGz/AllGrass
GzEffGrf is equivalent to the actual efficiency of grazers given the amount of grass
available in the system. It is a graphical function which takes as its input the ratio
of food needed compared to the grass available (GzGrass) and gives as an output
to the model the actual efficiency of grazers. As the amount of grass in the system
increases the efficiency of Grazers increases. It is assumed that at the highest
level of grass efficiency can no longer increase having reached its maximum and
by the same token that the greatest decrease will be in the center with efficiency
loss tailing off at lowest levels of grass. Therefore the graph is sigmoid in shape it
is shown in Figure 27:
= GRAPH (GzGrass)
GzEfficiency is equivalent to the realized efficiency of grazers it is the setting for optimal
efficiency multiplied by the actual efficiency of grazers;
= SetEffGz * GzEffGrf
100
Fig. 27. GzEffGr. Actual efficiency of Grazers given the amount of grass available in the system
Three herbivore model.
GzEffGrf=GRAPH(GzGrass)
(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, (1.190); (0.995,
1.010); (0.996, 0.853); (0.995, 0.725); (0.998, 0.620); (0.999, 0.538); (1.000, 0.515)
101
GrasNeedGz is equivalent to amount of grass taken out of the system by grazers it is the
realized efficiency of grazers times the efficiency of grazers;
= Grazers * GzEfficiency
(1–MxTreePerC) is equivalent to the amount of grass needed by MixedFeeders It is the
remainder of the diet once the fraction for trees has been set. It is therefore the
percentage of trees in the mixed feeder diet subtracted from one (1)
GrassNeedMx is equivalent to the grass needed by mixed feeders. It is the number of
mixed feeders times the efficiency of MixedFeeders times the amount of grass
needed (1–MxTreePerC);
= MixedFeeders * (MxEfficiency * (1–MxTreePerC)
GrassNeed is equivalent to the amount of grass taken out of the system by herbivores. It
is the sum of grass needed by grazers and mixed feeders;
= GrassNeedGz + GrassNeedMx)
DRateHi is the non- herbivore related death rate. It is arbitrarily assigned a value of 0.01
OutHi is equivalent to the outflow from the stock of high quality grass. The equation has
a switch that evaluates whether there is grass left over after herbivores have eaten.
If there is only sufficient grass to feed herbivores then the non herbivore death
rate does not apply if there is grass then it applies the non- herbivore death rate to
the remaining grass;
102
= If GrassHigh < GrassNeed then GrassHigh else GrassNeed + ((GrassHigh –
GrassNeed) * DRateHi)
HiLoRate is equivalent to the rate at which high quality grass passes into low quality
grass = 0.66 (arbitrarily assigned)
HiToLo is the amount of grass that passes from high quality grass to low quality grass. It
is the amount of high quality grass multiplied by the rate at which it passed into
low quality grass;
= GrassHigh * HiLoRate
Low Quality Grass
GrassLow is equivalent to the stock of low quality grass which is equivalent to the
amount of low quality grass present at any given time that can be supported by the
low quality grass present at that time;
GrassLow (t) = GrassLow (t – dt) + (HiToLo – OutLo) * dt
DrateLo is the non-herbivore death rate of GrassLow. It is arbitrarily assigned a rate of
0.1.
LoGRatio is equivalent to the amount of increased impact, which occurs when herbivores
are forced to eat low quality grass. This reflects the need for herbivores to kill
more biomass of low quality grass to get the same nourishment. =1.5 (arbitrarily
assigned).
103
AmtLo is the amount of grass needed over the amount supplied by GrassHigh multiplied
by the amount of increased impact (LoGRatio);
= GrassNeed–GrassHigh * LoGRatio
EatLo evaluates the need for low quality grass to be removed from the system. If there is
enough high quality grass then it returns zero to the model, if not it calls AmtLo;
= If GrassNeed >GrassHigh THEN AmtLo ELSE 0
OutLo is equivalent to the outflow from the stock of low quality grass. It removes the
grass eaten from the system and applies the non- herbivore death rate to the
remaining grass;
= EatLo + ((GrassLow–EatLo) * DRateLo)
Herbivores
Browsers, Grazers and Mixed Feeders
Part of the redesign of this step is to replace herbivores of the Step 1 & 2 models with
browsers, grazers and mixed feeders.
Herbivores is the sum of all herbivore stocks;
= Browsers + Grazers + MixedFeeders
Browsers
Browsers are similar in structure to Herbivores from the base model.
104
Browsers the stock of browsers which is equivalent to the number of browsers present at
any given time, expressed in animal units;
= Browsers (t – dt) + (InBz – OutBz) * dt
BzBirthRate is the birth rate of browsers it is equal to 0.6.
BzDensity is equivalent to the density of Browsers;
= Browsers/Area
BzDensityEffectGrf is a graphical function that reflects the inability of Browsers to find a
mate and continue reproducing when the density of browsers (BzDensity ) drops
too low. The graph of the function is shown in Figure 28
BzBirth is the birth rate of browsers (BzBirthRate) multiplied by the density effect graph
(BzDensityEffectGrf)
(1–Browsers/AllTrees) is equivalent to the limit to herbivore population growth imposed
by the carrying capacity of the continent for trees; which is equivalent to the
maximum fraction of browsers that can be added to the continental stock of
browsers; expressed in animal units.
InBz is equivalent to the number of Browsers added to Browsers stock per unit time;
= Browsers*(BzBirth*(1-Browsers/AllTrees))
BzHuntGrfCrn is equivalent to the rate at which carnivores kill Browsers. The input to
the graph (BzDensity) (shown on the X-axis) determines what value of
BzHuntGrfCrn (shown on the Y-axis) will be returned as output to the model. As
105
Fig. 28. Effect of Browser density, as Browser density declines Browser birth function drops
toward zero. Three herbivore model.
BzDensityEffectGrf = GRAPH(BzDensity)
(0.00, 0.00), (0.004, 0.00), (0.008, 0.03), (0.012, 0.2), (0.016, 0.75), (0.02, 0.94), (0.024,
1.00), (0.028, 1.00), (0.032, 1.00), (0.036, 1.00), (0.04, 1.00)
106
Browser density increases Carnivores are able to kill more Browsers. The graph
of this function is shown in Figure 29
= GRAPH (BzDensity)
FoodNeedCrn is the amount of food needed per Carnivore it is set at 20 lbs. per pound of
Carnivore per year (the Cat House, 1995)
BzHuntingCrn is equivalent to the hunting pressure per unit of carnivores. It is a factor of
how much food is required by carnivores and the density of Browsers
= BzHuntGrfCrn * FoodNeedCrn
PrCBz is equivalent to the percentage of Browsers in the herbivore population. It is the
number of Browsers divided by the number of herbivores
= Browsers/Herbivores
PrCCrnBz is equivalent to the percentage of carnivores hunting devoted to Browsers. It is
a function of the percentage of Browsers in the herbivore population multiplied by
the number of carnivores.
= Carnivores * PrCBz
BzHCrn is equivalent to the number of units of Browsers killed by carnivores
= BzHuntingCrn * PrCCrnBz
BzHuntGrfHs is equivalent to the rate at which Hsapiens kill Browsers. The input to the
graph (BzDensity) (shown on the X-axis) determines what value of BzHuntGrfHs
(shown on the Y-axis) will be returned as output to the model. As browser density
107
Fig. 29. The rate at which Carnivores kill Browsers. Three herbivore model.
BzHuntGrfCrn = GRAPH(GzDensity)
(0.00, -0.001), (0.02, 0.000), (0.04, 0.030), (0.06, 0.061), (0.08, 0.082), (0.10, 0.105),
(0.12, 0.122), (0.14, 0.139), (0.16, 0.160), (0.18, 0.178), (0.20, 0.196)
108
increases Hsapiens are able to kill more Browsers. The graph of this function is
shown in Figure 30
= GRAPH (BzDensity)
FoodNeedHs is the amount of food needed per pound of Hsapiens per year. It is set at 10
which is half what an obligate carnivore needs.
BzHuntingHs is equivalent to the hunting pressure per unit of Hsapiens. It is a factor of
how much food is required by Hsapiens and the density of Browsers
= BzHuntGrfHs * FoodNeedHs
PrCHsBz is equivalent to the percentage of Hsapiens hunting devoted to Browsers. It is a
function of the percentage of Browsers in the herbivore population multiplied by
the number of Hsapiens.
= Hsapiens * PrCBz
BzHHs is equivalent to the number of units of Browsers killed by Hsapiens
= BzHuntingHs * PrCHsBz
(BzHHs + BzHCrn) is the sum of hunting demand on Browsers from Carnivores and
Hsapiens
HuntDelay is the amount of time it takes for Carnivores and H. sapiens to switch prey. It
is an arbitrarily assigned value of 1.5 units of time.
HuntBz is hunting demand delayed by the amount specified in HuntDelay.
=delay((BzHHs + BzHCrn),HuntDelay)
109
Fig. 30. The rate at which Hsapiens kill Browsers. Three herbivore model.
BzHuntGrfHs = GRAPH(BzDensity)
(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18,
0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)
110
DeathRateBz is the non-hunting death rate of browsers and is arbitrarily assigned a rate of
0.4.
OutBz is equivalent to the loss of Browsers from the stock of Browsers per unit time;
= (Browsers * DeathRateBz) + HuntBz
Grazers
Grazers the stock of Grazers which is equivalent to the number of Grazers present at any
given time, expressed in animal units
= Grazers(t – dt) + (InGz – OutGz) * dt
GzEffGrf is equivalent to the actual efficiency of Grazers given the amount of grass
available in the system. It is a graphical function which takes as its input the ratio
of food needed compared to the grass available (GzGrass) and gives as an output
to the model the actual efficiency of Grazers. The graph is shown in Figure 31
GzEffGrf = GzGrass
GzBirthGrf is equivalent to the birth rate of Grazers. The input to the graph
(GzEfficiency) (shown on the X-axis) determines what value of GzBirthGrf
(shown on the Y-axis) will be returned as output to the model. As the efficiency
of Grazers increases, the birth rate of Grazers (GzBirthGrf) increases linearly.
The graph of this function is shown in Figure 32
= GRAPH (GzEfficiency)
111
Fig. 31. Actual efficiency of Grazers given the amount of grass available in the system. Three
herbivore model.
GzEffGrf = GRAPH(GzGrass)
(0.99, 2.00), (0.991, 1.86), (0.992, 1.66), (0.993, 1.42), (0.994, 1.19), (0.995, 1.01),
(0.996, 0.853), (0.997, 0.725), (0.998, 0.62), (0.999, 0.538), (1.00, 0.515)
112
Fig. 32. The birth rate of Grazers. Three herbivore model
GzBirthGrf = GRAPH(GzEfficiency)
(0.6, 1.000); (0.69, 0.906); (0.78, 0.816); (0.87, 0.730); (0.96, 0.640); (1.05, 0.550);
(1.24, 0.460); (1.23, 0.370); (1.32, 0.276); (1.41, 0.190); (1.5, 0.100)
113
GzDensity is equivalent to the density of Grazers;
= Grazers/Area
GzDensityEffectGrf is a graphical function that reflects the inability of Grazers to find a
mate and continue reproducing when the density of Grazers drops too low. The
graph of the function is shown in Figure 33
GzBirth is the birth rate of Grazers (GzBirthGrf ) multiplied by the GzDensityEffectGrf
(1–Grazers/(AllGrass) is equivalent to the limit to grazer population growth imposed by
the carrying capacity of the continent for plants; which is equivalent to the
maximum fraction of Grazers that can be added to the continental stock of
Grazers; expressed in animal units.
InGz is equivalent to the number of Grazers added to Grazers stock per unit time;
= (Grazers * GzBirth) * (1–Grazers/(AllGrass))
GzHuntGrfCrn is equivalent to the rate at which carnivores kill Grazers. The input to the
graph (GzDensity) (shown on the X-axis) determines what value of
GzHuntGrfCrn (shown on the Y-axis) will be returned as output to the model. As
Grazer density increases Carnivores are able to kill more Grazers. The graph of
this function is shown in Figure 34
= GRAPH (GzDensity)
114
Fig. 33. Effect of Grazer density, as Grazer density declines Grazer birth function drops toward
zero. Three herbivore model
GzDensityEffectGrf = GRAPH(GzDensity)
(0.00, 0.00), (0.004, 0.00), (0.008, 0.03), (0.012, 0.2), (0.016, 0.75), (0.02, 0.94), (0.024,
1.00), (0.028, 1.00), (0.032, 1.00), (0.036, 1.00), (0.04, 1.00)
115
Fig. 34. The rate at which Carnivores kill Grazers. Three herbivore model.
GzHuntGrfCrn = GRAPH(GzDensity)
(0.00, -0.001), (0.02, 0.000), (0.04, 0.030), (0.06, 0.061), (0.08, 0.082), (0.10, 0.105),
(0.12, 0.122), (0.14, 0.139), (0.16, 0.160), (0.18, 0.178), (0.20, 0.196)
116
GzHuntingCrn is equivalent to the hunting pressure per unit of carnivores. It is a factor of
how much food is required by carnivores and the density of Grazers
= GzHuntGrfCrn * FoodNeedCrn
PrCGz is equivalent to the percentage of Grazers in the herbivore population. It is the
number of Grazers divided by the number of herbivores
= Grazers/Herbivores
PrCCrnGz is equivalent to the percentage of carnivores hunting devoted to Grazers. It is
a function of the percentage of Grazers in the herbivore population multiplied by
the number of carnivores.
= Carnivores * PrCGz
GzHCrn is equivalent to the number of units of Grazers killed by carnivores
= GzHuntingCrn * PrCCrnGz
GzHuntGrfHs is equivalent to the rate at which Hsapiens kill Grazers. The input to the
graph (GzDensity) (shown on the X-axis) determines what value of GzHuntGrfHs
(shown on the Y-axis) will be returned as output to the model. As Grazer density
increases Hsapiens are able to kill more Grazers. The graph of this function is
shown in Figure 35
= GRAPH (GzDensity)
117
Fig. 35. The rate at which Hsapiens kill Grazers. Three herbivore model.
GzHuntGrfHs = GRAPH(GzDensity)
(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18,
0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)
118
GzHuntingHs is equivalent to the hunting pressure per unit of Hsapiens. It is a factor of
how much food is required by Hsapiens and the density of Grazers
= GzHuntGrfHs * FoodNeedHs
PrCHsGz is equivalent to the percentage of Hsapiens hunting devoted to Grazers. It is a
function of the percentage of Grazers in the herbivore population multiplied by
the number of Hsapiens.
= Hsapiens * PrCGz
GzHHs is equivalent to the number of units of Grazers killed by Hsapiens
= GzHuntingHs * PrCHsGz
(GzHHs + GzHCrn) is the sum of hunting demand on Grazers from Carnivores and
Hsapiens
HuntGz is hunting demand delayed by the amount specified in HuntDelay.
=delay((GzHHs + GzHCrn),HuntDelay)
GzDeathRate is the non-hunting death rate of grazers and is arbitrarily assigned a rate of
0.25.
OutGz is equivalent to the loss of Grazers from the stock of Grazers per unit time;
= (Grazers * GzDeathRate) + HuntGz
119
Mixed Feeders
Mixed feeders are dependent on both Plants sectors for different parts of their nutritional
mix. According to Wing & Buss (1970) and Anderson & Walker (1984), elephants do not
reproduce when they are deprived of browse and they need grass to give them sufficient
calories to sustain life.
MixedFeeders the stock of mixed feeders which is equivalent to the number of mixed
feeders present at any given time, expressed in animal units
= MixedFeeders(t – dt) + (InMx – OutMx) * dt
MxBirthGrf is equivalent to the birth rate of MixedFeeders. The input to the graph
(TreeFactorMx) (shown on the X-axis) determines what value of MxBirthGrf
(shown on the Y-axis) will be returned as output to the model. As TreeFactorMx
increases the birth rate of MixedFeeders (MxBirthGrf) increases linearly.. The
graph of this function is shown in Figure 36
MxDensity is equivalent to the density of MixedFeeders
= MixedFeeders/Area
MxDensityEffectGrf is a graphical function that reflects the inability of MixedFeeders to
find a mate and continue reproducing when the density of MixedFeeders drops
too low. The graph of the function is shown in Figure 37
MxBirth is the birth rate of MixedFeeders (MxBirthGrf ) multiplied by the
MxDensityEffectGrf
120
Fig. 36. The birth rate of MixedFeeders. Three herbivore model.
MxBirthGrf=GRAPH(TreeFactorMx)
(0.950, 0.000); (0.955, 0.070); (0.960, 0.137); (0.965, 0.207); (0.970, 0.277); (0.975,
0.347); (0.980, 0.417); (0.985, 0.483); (0.990, 0.557); (0.995, 0.626); (1.000, 0.697)
121
Fig. 37. Effect of MixedFeeder density, as MixedFeeder density declines MixedFeeder birth
function drops toward zero. Three herbivore model.
MxDensityEffectGrf = GRAPH(MxDensity)
(0.00, 0.00), (0.004, 0.00), (0.008, 0.03), (0.012, 0.2), (0.016, 0.75), (0.02, 0.94), (0.024,
1.00), (0.028, 1.00), (0.032, 1.00), (0.036, 1.00), (0.04, 1.00)
122
InMx is equivalent to the number of MixedFeeders added to MixedFeeder stock per unit
time. It is the number of MixedFeeders multiplied by MxBirth.
= MixedFeeders*MxBirth
MxHuntGrfCrn is equivalent to the rate at which carnivores kill MixedFeeders. The input
to the graph (MxDensity) (shown on the X-axis) determines what value of
MxHuntGrfCrn (shown on the Y-axis) will be returned as output to the model. As
MixedFeeder density increases Carnivores are able to kill more MixedFeeders.
The graph is shown in Figure 38
= GRAPH (MxDensity)
MxHuntingCrn is equivalent to the hunting pressure per unit of carnivores. It is a factor
of how much food is required by carnivores and the density of MixedFeeders
= MxHuntGrfCrn * FoodNeedCrn
PrCMx is equivalent to the percentage of MixedFeeders in the herbivore population. It is
the number of MixedFeeders divided by the number of herbivores
= MixedFeeders/Herbivores
PrCCrnMx is equivalent to the percentage of carnivores hunting devoted to
MixedFeeders. It is a function of the percentage of MixedFeeders in the herbivore
population multiplied by the number of Carnivores.
= Carnivores * PrCMx
123
Fig. 38. The rate at which Carnivores kill MixedFeeders. Three herbivore model.
MxHuntGrfCrn = GRAPH(MxDensity)
(0.00, -0.001), (0.02, 0.000), (0.04, 0.030), (0.06, 0.061), (0.08, 0.082), (0.10, 0.105),
(0.12, 0.122), (0.14, 0.139), (0.16, 0.160), (0.18, 0.178), (0.20, 0.196)
124
MxHCrn is equivalent to the number of units of MixedFeeders killed by Carnivores
= MxHuntingCrn * PrCCrnMx
MxHuntGrfHs is equivalent to the rate at which Hsapiens kill MixedFeeders. The input to
the graph (MxDensity) (shown on the X-axis) determines what value of
MxHuntGrfHs (shown on the Y-axis) will be returned as output to the model. As
MixedFeeder density increases Hsapiens are able to kill more MixedFeeders. The
graph of this function is shown in Figure 39
= GRAPH (MxDensity)
MxHuntingHs is equivalent to the hunting pressure per unit of Hsapiens. It is a factor of
how much food is required by Hsapiens and the density of MixedFeeders
= MxHuntGrfHs * FoodNeedHs
PrCHsMx is equivalent to the percentage of Hsapiens hunting devoted to MixedFeeders.
It is a function of the percentage of MixedFeeders in the herbivore population
multiplied by the number of Hsapiens.
= Hsapiens * PrCMx
MxHHs is equivalent to the number of units of MixedFeeders killed by Hsapiens
= MxHuntingHs * PrCHsMx
(MxHHs + MxHCrn) is the sum of hunting demand on MixedFeeders from Carnivores
and Hsapiens
125
Fig. 39. The rate at which Hsapiens kill MixedFeeders. Three herbivore model.
MxHuntGrfHs = GRAPH(MxDensity)
(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18,
0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)
126
HuntMx is hunting demand delayed by the amount specified in HuntDelay.
=delay((MxHHs + MxHCrn),HuntDelay)
GrassFactorMx is the ratio of grass needed by MixedFeeders to all grass subtracted from
one (1)
= 1 – GrassNeedMx/AllGrass
DMx is equivalent to the death rate of mixed feeders according to the amount of grass in
the environment relative to the amount needed by mixed
feeders(GrassFactorMx). As GrassFactorMx declines the death rate increases
linearly as shown in Figure 40
= GRAPH (GrassFactorMx)
OutMx is the out flow from the stock of mixed feeders. It is MixedFeeders multiplied by
the non-hunting death (DMx) plus the death from hunting (HuntMx).
Carnivores
CarnivoreDenisty is Carnivores per unit of area
= Carnivores/Area
CrnDensityGrf is a function that modifies the rate that Hsapiens can hunt Carnivores
relative to their density. As Carnivore density declines the modifier declines. The
graph of the function is shown in Figure 41
= GRAPH(CarnivoreDensity)
127
Fig. 40. The death rate of MixedFeeders according to the amount of grass in the environment
relative to the amount needed. Three herbivore model.
MxDeathGrf = GRAPH(GrassFactorMx)
(0.950, 1.000); (0.955, 0.915); (0.960, 0.825); (0.965, 0.730); (0.970, 0.620); (0.975,
0.525); (0.980, 0.430); (0.985, 0.320); (0.990, 0.225); (0.995, 0.110); (1.000, 0.015)
128
Fig. 41. The rate Hsapiens hunts Carnivores relative to their density. Three herbivore model.
CrnDensityGrf = GRAPH(CrnDensity)
(0.0158, 0.435), (0.0211, 0.775), (0.0263, 0.94), (0.0316, 1), (0.0421, 1), (0.0526, 1), (0.0632, 1),
(0.075, 1), (0.08, 1), (0.0895, 1), (0.1, 1)
129
AmtHsKillsCrn is the rate that Hsapiens kills Carnivores the default setting is 0.025.
OutStepCrn is the amount that Hsapiens kills Carnivores multiplied by CrnDensityGrf
= Hsapiens*(CrnDensityGrf*AmtHsKillCrn)
Results of Step 3: Three-Herbivore Model
Graph of the Model
The first test of the model is to see that it reproduces its reference modes. The first
reference mode to be reproduced is the equilibrium situation of Step 1 of the base model.
The Graph shows that the model returns to equilibrium when perturbed by a one time
pulse reduction of Carnivore populations of 5%. This is an increase over the perturbation
of the Base model because the other sectors did not respond visibly to the smaller
perturbation. Figure 42shows that the Three-Herbivore Model does reproduce the
reference mode.
The second reference mode to be reproduced is the introduction of the second
predator (Hsapiens) – Overkill Mode. The graph is shown in Figure 43
In the Three-Herbivore model the introduction of Hsapiens causes very little
disturbance to any of the sectors. It produces less of an impact than it does in the base
model. Populations are 90% of starting values. Carnivore populations are reduced
relatively more than Herbivores but only a fraction of a percent, food for herbivores, like
in the base model, increases slightly to 101% of its starting value
130
Fig. 42. Equilibrium mode graph. Three herbivore model
131
Fig. 43. - Second predator (overkill) mode, aggregated view. Three herbivore model.
132
The third reference mode is the second-order overkill mode, where Hsapiens hunts
carnivores. This results in a major crash of Herbivore and Carnivore populations.
Hsapiens population levels off sooner than in the Second Predator (Overkill) mode
shown in Figure 44.
To see why these results were obtained it is useful to look at herbivore populations
shown in Figure 45.
Browsers and mixed feeders expand and crash. Grazers, initially have a population
slump, as they bear all the pressure of predation by both Carnivores and Hsapiens. As
Hsapiens and Carnivore populations stabilize, grazers rebound, and stabilize. To
understand the dynamics of the herbivore populations it is necessary to look at Plants
shown in Figure 46.
Browsers and mixed feeders eat trees faster than trees can recruit, resulting in a
complete crash of small trees, followed by near extinction of large trees as well. The dip
in grazer populations after the extinction of browsers and mixed feeders allows the two
grass sectors to boom until grazer populations equilibrate.
133
Fig. 44. Second-order predation, aggregated view. Three herbivore model
134
Fig. 45. Second-order predation, herbivores Three herbivore model
135
Fig. 46. Second-order predation, plants Three herbivore model
136
Step 4 – Four Herbivores (Browsers, Ruminant Grazers, Non-ruminant
Grazers and Mixed Feeders)
Overview
Step four will establish, whether under step three conditions, ruminants are competitively
favored over monogastric herbivores. For this it will be necessary to partition grazers into
ruminants and non-ruminants and modify the equations which specify the amount and
kind of vegetation necessary for ruminants and monogastrics to extract sufficient energy
to sustain life and reproduce. This should result in a reduction of monogastrics relative to
ruminants. All conventions and definitions, not discussed, remain as previously specified.
Diagram of the Model
The only difference between the Three-Herbivore model and the– Four-Herbivore model
is the addition of equations describing the difference in grass consumption between
ruminants and non-ruminants. Ruminants are able to eat low quality grass once their
portion of high quality grass is depleted. shown in Figure 47
An illustration of the change in Grazers is shown in Figure 48.
Conventions and definition of terms used in Step 4
Array an array is a set of variables which are conceptually arranged in rows and columns.
The partitioning of grazers into ruminants and non-ruminants is done through an
array. Thus all the variables, which pertained to grazers in the Three-Herbivore
137
Fig. 47. Grass, diagram. Four-herbivore model.
Key as in Base Model diagram.
138
Fig. 48. Grazers diagram. Four-herbivore model.
Key as in Base Model diagram.
139
Model now have values apportioned by digestion type and levels accumulate by digestion
type. These are shown in the documentation as Variable [Digestion] to signify that it applies
to both ruminants and non- ruminants, Variable [Ruminant] to signify that it applies to
ruminants and Variable [NonRuminant] to signify that it applies to non- ruminants. The
concept is illustrated in Table 4 below
Table – 4 – Array Illustration
Digestion Type Ruminant Non- Ruminant
Population initialized at equilibrium equilibrium
SetEffGz 0.9 0.8
DGz 0.4 0.4
Grazers 1043.13K 747.75K
ARRAYSUM (Variable [ * ]) is the sum of the values for an arrayed variable for using the
example above the ARRAYSUM (Grazers [ * ]) is 1790.88K.
Continent
Herbivore Grass Eating Preferences
Ruminant grazers prefer high quality grass to low quality grass. If the demand for high
quality grass exceeds the stock of high quality grass, low quality grass is eaten. More low
quality grass is necessary to provide the same level of nourishment as high quality grass.
Thus each unit of nutritional demand on GrassLow has a 1.5 unit impact.
140
For the purposes of the model it is assumed that non-ruminant grazers eat only high
quality grass.
With the partitioning of grazers into ruminants and non-ruminants it is necessary to
apportion grass amongst them. It is assumed that grass is eaten in proportion to the
number of animals in the population.
SetEffGz [Ruminant] is equivalent to the setting for optimal ruminant Grazer efficiency;
it is arbitrarily assigned a value of 0.8
SetEffGz [NonRuminant] is equivalent to the setting for optimal non-ruminant grazer
efficiency; it is arbitrarily assigned a value of 0.9
(Grazers [Ruminant] * SetEffGz [Ruminant]) is equivalent to the amount of food needed
to support ruminant grazers. It is the population of ruminant grazers multiplied by
setting for optimal ruminant grazer efficiency.
(Grazers[NonRuminant] * SetEffGz [NonRuminant]) is equivalent to the amount of food
needed to support non-ruminant grazers. It is the population of non-ruminant
grazers multiplied by setting for optimal non-ruminant grazer efficiency.
FoodNeed is equivalent to the amount of food necessary to support the population of
grazers; it is the sum of the food needed by each of the grazer population;
= (Grazers [Ruminant] * SetEffGz [Ruminant]) + (Grazers[NonRuminant] *
SetEffGz [NonRuminant])
141
GzEffGrf [Ruminant] is equivalent to the actual efficiency of ruminant grazers given the
amount of grass available in the system. It is a graphical function which takes as
its input the ratio of food needed compared to the grass available (GzGrass) and
gives as an output to the model the actual efficiency of ruminant grazers. It is
assumed that efficiency will only increase to a certain extent so efficiency will
increase at a slower rate as there becomes an excess of grass by the same token
there will be a slowing in the rate of decline in the decrease of efficiency as grass
decreases. If the relationship between grass and grazer efficiency is graphed the
curve will be slightly sigmoid. The center of the graph will have the steepest slope
because it is the area of greatest change whereas the ends of the graph will be
tapered since the rate of change is less. Therefore the graph is sigmoid in shape it
is shown in Figure 49
GzEffGrf [Ruminant] = GzGrass
GzEffGrf [NonRuminant] is equivalent to the actual efficiency of non-ruminant grazers
given the amount of grass available in the system. It is a graphical function which
takes as its input the ratio of food needed compared to the grass available
(GzGrass) and gives as an output to the model the actual efficiency of non-
ruminant grazers. As the amount of grass in the system increases the efficiency of
non-ruminant grazers increases. As with ruminants the center of the graph has the
steepest slope and is the area of greatest change. The following are assumed.
Firstly that non-ruminants are less efficient, through the middle range, than are
ruminants. Secondly that non-ruminants will benefit proportionally more than
142
Fig. 49. Actual efficiency of ruminant grazers (Grazers[Ruminant]) given the amount of grass
available in the system. Four-herbivore model.
GzEffGrf[Ruminant]=GRAPH(GzGrass)
(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, 1.190); (0.995, 1.010);
(0.996, 0.853); (0.997, 0.725); (0.998, 0.620); (0.999, 0.538); (1.000, 0.515)
143
ruminants from an increase in grass, so the top of the graph does not level off.
And finally that as grass becomes more scarce efficiency will taper off more
slowly than it does for ruminants. Thus the curve for non-ruminant grazer
efficiency is slightly concave it is shown in Figure 50
GzEffGrf [NonRuminant] = GzGrass
GzEfficiency [Digestion] is equivalent to the realized efficiency of each of the grazers it
is the setting for optimal efficiency of each of the grazers multiplied by the actual
efficiency of each of the grazers;
= SetEffGz [Digestion] * GzEffGrf [Digestion]
GrasNeedGz [Digestion] is equivalent to amount of grass taken out of the system by each
of the grazers it is the realized efficiency of each of the grazers times the
efficiency of each of the grazers;
= Grazers [Digestion] * GzEfficiency [Digestion]
ARRAYSUM (GrasNeedGz [ * ]) is equivalent to the sum of the grass taken out of the
system by both grazers.
GrassNeed is equivalent to the grass taken out of the system by grazers and mixed
feeders.
= ARRAYSUM (GrasNeedGz [ * ]) + GrassNeedMx
144
Fig. 50. Actual efficiency of non–ruminant grazers(Grazers[NonRuminant]) given the amount of
grass available in the system. Four herbivore model
GzEffGrf[NonRuminant]=GRAPH(GzGrass)
(0.990, 2.000); (0.991, 1.603); (0.992, 1.355); (0.993, 1.168); (0.994, 1.025); (0.995, 0.890);
(0.996, 0.785); (0.997, 0.688); (0.998, 0.613); (0.999, 0.560); (1.000, 0.523)
145
SumGz is equivalent to the entire population of grazers. It is the sum of ruminant and
non-ruminant grazers
= ARRAYSUM (Grazers [ * ])
RatiosGz [Digestion] is the ratio of each of the grazers and is found by dividing the
number of grazers (SumGz) by the number of each of the grazers (Grazers
[Digestion])
= Grazers [Digestion]/SumGz
AvailHi [Digestion] is equivalent to the amount of high quality grass allotted to each of
the grazers in the population of grazers. It is the amount of high quality grass
(GrassHigh) times the ratio of each of the grazers to the entire population of
grazers RatiosGz [Digestion]
= GrassHigh * RatiosGz [Digestion]
LoGRatio is equivalent to the amount of increased impact on big trees when ruminant
grazers are forced to eat a less preferred food, low quality grass (GrassLow). This
reflects the need for herbivores to kill more biomass of low quality grass to obtain
the same nourishment. It is equal to 1.5 (arbitrarily assigned).
AmtLo is equivalent to the amount of grass taken out of the low quality grass stock by
ruminant grazers. It multiplies the amount of grass demand not filled by high
quality grass by the amount of increased impact (LoGRatio). It is determined by
subtracting the amount of high quality grass available to ruminant grazers from
the amount of grass demand by ruminant grazers (GrasNeedGz [Ruminant]) and
146
multiplying the remainder by the amount of increased impact (LoGRatio).
= (GrasNeedGz [Ruminant] –AvailHi [Ruminant]) * LoGRatio
Herbivores
Herbivores is equivalent to the entire population of herbivores. It is the sum of browsers,
mixed feeders and the sum of the array of grazers, ruminant and non- ruminant.
Herbivores = Browsers + MixedFeeders + ARRAYSUM (Grazers [ * ])
Grazers
Grazers [Ruminant] is the stock of Grazers which is equivalent to the number of
ruminant grazers present at any given time, expressed in animal units.
= Grazers [Ruminant] (t – dt) + (InGz [Ruminant] – OutGz [Ruminant]) * dt
GzBirthGrf [Ruminant] is equivalent to the birth rate of ruminants. It is a graphical
function that takes ruminant grazer efficiency GzEfficiency [Ruminant] as its
input and returns the birth rate of ruminant grazers GzBirthGrf [Ruminant] as the
output. The slope of the graph of the relationship between ruminant grazer
efficiency and ruminant grazer birth is very flat as ruminants are assumed to be
able to reproduce under most conditions. The shape of the graph is slightly
sigmoid – As efficiency decreases the birth rate decreases. The graph is shown in
Figure 51
= GzEfficiency [Ruminant]
147
Fig. 51. The birth rate of ruminants Grazers[Ruminant] Four herbivore model.
GzBirthGrf[Ruminant]=GRAPH(GzEfficiency[Ruminant])
(0.600, 0.591); (0.690, 0.569); (0.780, 0.559); (0.870, 0.551); (0.960, 0.548); (1.050, 0.545);
(1.140, 0.539); (1.230, 0.535); (1.320, 0.528); (1.410, 0.521); (1.500, 0.508)
148
GzDensity[Ruminant] is the density of Grazers[Ruminant]
= Grazers[Ruminant]/Area
GzDensityEffectGrf[Ruminant] is a graphical function that reflects the inability of
Grazers[Ruminant] to find a mate and continue reproducing when the density of
grazers (GzDensity[Ruminant]) drops too low. The graph of the function is shown
in Figure 52
GzBirth[Ruminant] is the birth rate of ruminant as found in the birth rate graph
(GzBirthGrf[Ruminant]) modified by the effect of density of ruminants
(GzDensityEffectGrf[Ruminant])
= GzDensityEffectGrf[Ruminant]*GzBirthGrf[Ruminant]
(1–Grazers [Ruminant]/(AllGrass)) is equivalent to the limit to ruminant grazer
population growth imposed by the carrying capacity of the continent for plants;
which is equivalent to the maximum fraction of ruminant grazers that can be
added to the continental stock of ruminant grazers Grazers [Ruminant] ;
expressed in animal units.
InGz [Ruminant] is equivalent to the number of ruminant grazers Grazers [Ruminant]
added to ruminant grazers Grazers [Ruminant] stock per unit time;
=Grazers [Ruminant] * (GzBirth [Ruminant]) * (1–Grazers
[Ruminant]/(AllGrass))
149
Fig. 52. Effect of Grazer[Ruminant] density, as Grazers[Ruminant] density declines
Grazers[Ruminant] birth function drops toward zero. Four-herbivore model.
GzDensityEffectGrf[Ruminant] = GRAPH(GzDensity[Ruminant])
(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.215), (0.0136, 0.825),
(0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00), (0.0273, 1.00), (0.03, 1.00)
150
GzHuntGrfCrn [Ruminant] is equivalent to the rate at which carnivores kill ruminant
grazers Grazers [Ruminant] . The input to the graph (GzDensity [Ruminant])
(shown on the X-axis) determines what value of GzHuntGrfCrn [Ruminant]
(shown on the Y-axis) will be returned as output to the model. As ruminant grazer
(Grazers [Ruminant] ) density increases Carnivores are able to kill more ruminant
grazers (Grazers [Ruminant]). The graph of this function is shown in Figure 53=
GRAPH (GzDensity [Ruminant])
GzHuntingCrn [Ruminant] is equivalent to the hunting pressure per unit of carnivores. It
is a factor of how much food is required by carnivores and the density of ruminant
grazers
= GzHuntGrfCrn [Ruminant] * FoodNeedCrn
PrCGz [Ruminant] is equivalent to the percentage of ruminant grazers (Grazers
[Ruminant]) in the herbivore population. It is the number of ruminant grazers
(Grazers [Ruminant]) divided by the number of herbivores
= Grazers [Ruminant]/Herbivores
PrCCrnGz [Ruminant] is equivalent to the percentage of carnivores hunting devoted to
ruminant grazers (Grazers [Ruminant]). It is a function of the percentage of
ruminant grazers (Grazers [Ruminant]) in the herbivore population multiplied by
the number of carnivores.
= Carnivores * PrCGz [Ruminant]
151
Fig. 53. The rate at which Carnivores kill ruminant grazers (Grazers[Ruminant]). Four
Herbivore Model.
GzHuntGrfCrn[Ruminant] = GRAPH(GzDensity[Ruminant])
(0.00, 0.000); (0.02, 0.020); (0.04, 0.040); (0.06, 0.061); (0.08, 0.081); (0.10, 0.099); (0.12,
0.120); (1.40, 0.139); (0.16, 0.160); (0.18, 0.178); (0.20, 0.199)
152
GzHCrn [Ruminant] is equivalent to the number of units of ruminant grazers (Grazers
[Ruminant]) killed by carnivores
= GzHuntingCrn [Ruminant] * PrCCrnGz [Ruminant]
GzHuntGrfHs [Ruminant] is equivalent to the rate at which Hsapiens kill ruminant
grazers (Grazers [Ruminant]). The input to the graph (GzDensity [Ruminant])
(shown on the X-axis) determines what value of GzHuntGrfHs [Ruminant]
(shown on the Y-axis) will be returned as output to the model. As ruminant grazer
(Grazers [Ruminant] ) density increases Hsapiens are able to kill more ruminant
grazers (Grazers [Ruminant]). The graph of this function is shown in Figure 54
= GRAPH (GzDensity [Ruminant])
GzHuntingHs [Ruminant] is equivalent to the hunting pressure per unit of Hsapiens. It is
a factor of how much food is required by Hsapiens and the density of ruminant
grazers (Grazers [Ruminant])
= GzHuntGrfHs [Ruminant] * FoodNeedHs
PrCHsGz [Ruminant] is equivalent to the percentage of Hsapiens hunting devoted to
ruminant grazers (Grazers [Ruminant]). It is a function of the percentage of
ruminant grazers (Grazers [Ruminant]) in the herbivore population multiplied by
the number of Hsapiens.
= Hsapiens * PrCGz [Ruminant]
153
Fig. 54. The rate at which Hsapiens kill ruminant grazers (Grazers[Ruminant]). Four Herbivore
Model.
GzHuntGrfHs[Ruminant] =GRAPH(GzDensity[Ruminant])
(0, 0); (0.03, 0.008); (0.06, 0.021); (0.09, 0.039); (0.12, 0.062); (0.15, 0.095); (0.18, 0.127);
(0.21, 0.159); (0.24, 0.183); (0.27, 0.193); (0.3, 0.199)
154
GzHHs [Ruminant] is equivalent to the number of units of ruminant grazers (Grazers
[Ruminant]) killed by Hsapiens
= GzHuntingHs [Ruminant] * PrCHsGz [Ruminant]
(GzHHs [Ruminant] + GzHCrn [Ruminant]) is the sum of hunting demand on ruminant
grazers (Grazers [Ruminant]) from Carnivores and Hsapiens
HuntGz [Ruminant] is hunting demand delayed by the amount specified in HuntDelay.
=delay((GzHHs [Ruminant] + GzHCrn [Ruminant]),HuntDelay)
DGz [Ruminant] is the non- hunting death rate of ruminant grazers (Grazers [Ruminant])
and is arbitrarily assigned a rate of 0.25.
OutGz [Ruminant] is equivalent to the loss of ruminant grazers (Grazers [Ruminant])
from the stock of ruminant grazers (Grazers [Ruminant]) per unit time;
= Grazers [Ruminant] * (DGz [Ruminant]) + HuntGz [Ruminant]
Non-ruminant grazers
BGz [NonRuminant] is equivalent to the birth rate of non- ruminants. It is a graphical
function that takes non-ruminant grazer efficiency GzEfficiency [NonRuminant]
as its input and returns the birth rate of non-ruminant grazers BGz [NonRuminant]
as the output. The slope of the graph of the relationship between non-ruminant
grazer efficiency and non-ruminant grazer birth is steep as non-ruminants are
assumed to be very sensitive to environmental conditions. The shape of the curve
155
is linear – As efficiency decreases the birth rate decreases, as shown in Figure 55
= GzEfficiency [NonRuminant]
GzDensity[NonRuminant] is the density of Grazers[NonRuminant]
= Grazers[NonRuminant]/Area
GzDensityEffectGrf[NonRuminant] is a graphical function that reflects the inability of
Grazers[NonRuminant] to find a mate and continue reproducing when the density
of grazers (GzDensity[NonRuminant]) drops too low. The graph of the function is
shown in Figure 56
GzBirth[NonRuminant] is the birth rate of NonRuminant as found in the birth rate graph
(GzBirthGrf[NonRuminant]) modified by the effect of density of non-ruminants
(GzDensityEffectGrf[NonRuminant])
= GzDensityEffectGrf[NonRuminant]*GzBirthGrf[NonRuminant]
(1–Grazers[NonRuminant]/(AllGrass)) is equivalent to the limit to non-ruminant grazer
population growth imposed by the carrying capacity of the continent for plants;
which is equivalent to the maximum fraction of non-ruminant grazers that can be
added to the stock of non-ruminant grazers Grazers[NonRuminant] ; expressed in
animal units.
InGz [NonRuminant] is equivalent to the number of non-ruminant grazers
Grazers[NonRuminant] added to non-ruminant grazers Grazers[NonRuminant]
stock per unit time;
156
Fig. 55. The birth rate of non-ruminants (Grazers[NonRuminant]). Four-herbivore model.
GzBirthGrf[NonRuminant]=GRAPH(GzEfficiency[NonRuminant])
(0.825, 0.906), (0.900, 0.816), (0.975, 0.730), (1.050, 0.640), (1.250, 0.550), (1.200, 0.460),
(1.275, 0.370), (1.350, 0.276), (1.425, 0.190), (1.500, 0.100 )
157
Fig. 56. Effect of non-ruminant density, as Grazers[NonRuminant] density declines
Grazers[NonRuminant] birth function drops toward zero. Four-herbivore model.
GzDensityEffectGrf[NonRuminant] =GRAPH(GzDensity[NonRuminant])
(0, 0); (0.03, 0.008); (0.06, 0.021); (0.09, 0.039); (0.12, 0.062); (0.15, 0.095); (0.18, 0.127);
(0.21, 0.159); (0.24, 0.183); (0.27, 0.193); (0.3, 0.199)
158
=Grazers[NonRuminant] * (BirthGz [NonRuminant]) * (1–
Grazers[NonRuminant]/(AllGrass))
GzHuntGrfCrn [NonRuminant] is equivalent to the rate at which carnivores kill non-
ruminant grazers Grazers[NonRuminant] . The input to the graph (GzDensity
[NonRuminant]) (shown on the X-axis) determines what value of GzHuntGrfCrn
[NonRuminant] (shown on the Y-axis) will be returned as output to the model. As
non-ruminant grazer (Grazer [NonRuminant]) density increases Carnivores are
able to kill more non-ruminant grazers (Grazers[NonRuminant]). The graph of
this function is shown in Figure 57
= GRAPH (GzDensity [NonRuminant])
GzHuntingCrn [NonRuminant] is equivalent to the hunting pressure per unit of
carnivores. It is a factor of how much food is required by carnivores and the
density of non-ruminant grazers (Grazers[NonRuminant])
= GzHuntGrfCrn [NonRuminant] * FoodNeedCrn
PrCGz [NonRuminant] is equivalent to the percentage of non-ruminant grazers
(Grazers[NonRuminant]) in the herbivore population. It is the number of non-
ruminant grazers (Grazers[NonRuminant]) divided by the number of herbivores
= Grazers[NonRuminant]/Herbivores
PrCCrnGz [NonRuminant] is equivalent to the percentage of carnivores hunting devoted
to non-ruminant grazers (Grazers[NonRuminant]). It is a function of the
percentage of non-ruminant grazers (Grazers[NonRuminant]) in the herbivore
159
Fig. 57. The rate at which carnivores kill non-ruminant grazers (Grazers[NonRuminant]). Four-
herbivore model.
GzHuntGrfCrn[NonRuminant] = GRAPH(GzDensity[NonRuminant])
(0.00, 0.000); (0.02, 0.020); (0.04, 0.040); (0.06, 0.061); (0.08, 0.081); (0.10, 0.099); (0.12,
0.120); (1.40, 0.139); (0.16, 0.160); (0.18, 0.178); (0.20, 0.199)
160
population multiplied by the number of carnivores.
= Carnivores * PrCGz [NonRuminant]
GzHCrn [NonRuminant] is equivalent to the number of units of non-ruminant grazers
(Grazers[NonRuminant]) killed by carnivores.
= GzHuntingCrn [NonRuminant] * PrCCrnGz [NonRuminant]
GzHuntGrfHs [NonRuminant] is equivalent to the rate at which Hsapiens kill non-
ruminant grazers (Grazers[NonRuminant]). The input to the graph (GzDensity
[NonRuminant]) (shown on the X-axis) determines what value of GzHuntGrfHs
[NonRuminant] (shown on the Y-axis) will be returned as output to the model. As
non-ruminant grazer (Grazer [NonRuminant]) density increases Hsapiens are able
to kill more non-ruminant grazers (Grazers[NonRuminant]). The graph of this
function is shown in Figure 58
= GRAPH (GzDensity [NonRuminant])
GzHuntingHs [NonRuminant] is equivalent to the hunting pressure per unit of Hsapiens.
It is a factor of how much food is required by Hsapiens and the density of non-
ruminant grazers (Grazers[NonRuminant])
= GzHuntGrfHs [NonRuminant] * FoodNeedHs
PrCHsGz [NonRuminant] is equivalent to the percentage of Hsapiens hunting devoted to
non-ruminant grazers (Grazers[NonRuminant]). It is a function of the percentage
of non-ruminant grazers (Grazers[NonRuminant]) in the herbivore population
161
Fig. 58. The rate at which Hsapiens kill non-ruminant grazers (Grazers[NonRuminant]). Four-
herbivore model.
GzHuntGrfHs[NonRuminant] = GRAPH(GzDensity[NonRuminant])
(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18, 0.127), (0.21,
0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)
162
multiplied by the number of H. sapiens.
= Hsapiens * PrCGz [NonRuminant]
GzHHs [NonRuminant] is equivalent to the number of units of non-ruminant grazers
(Grazers[NonRuminant]) killed by Hsapiens
= GzHuntingHs [NonRuminant] * PrCHsGz [NonRuminant]
(GzHHs [NonRuminant] + GzHCrn [NonRuminant]) is the sum of hunting demand on
non-ruminant grazers (Grazers[NonRuminant]) from Carnivores and Hsapiens
HuntGz [NonRuminant] is hunting demand delayed by the amount specified in
HuntDelay.
=delay ((GzHHs [NonRuminant] + GzHCrn [NonRuminant]),HuntDelay)
DGz [NonRuminant] is the non-hunting death rate of non-ruminant grazers
(Grazers[NonRuminant]) and is arbitrarily assigned a rate of 0.25.
OutGz [NonRuminant] is equivalent to the loss of non-ruminant grazers
(Grazers[NonRuminant]) from the stock of non-ruminant grazers
(Grazers[NonRuminant]) per unit time;
= Grazers[NonRuminant] * (DGz [NonRuminant]) + HuntGz [NonRuminant]
163
Results of the– Four-Herbivore Model
Graph of the Model
The first test of the model is to see that it reproduces its reference modes. The first
reference mode to be reproduced is the equilibrium situation of Step 1 of the Base model.
The graph shows that the model returns to equilibrium when perturbed by a single pulse
reduction of carnivore populations of 3%. This is an increase over the perturbation of the
Base model because the other sectors did not respond visibly to the smaller perturbation.
Figure 59 shows that the– Four-Herbivore Model does reproduce the reference mode.
The second reference mode to be reproduced is the introduction of the second
predator (Hsapiens) – Overkill Mode. The graph is shown in Figure 60
In the– Four-Herbivore model the introduction of Hsapiens causes very little
disturbance to any of the sectors – less than in the base model. Populations are 90% of
starting values. Carnivore populations are reduced relatively more than herbivores but
only a fraction of a percent. Plants, like in the base model, increases slightly to 101% of
its starting value
The third reference mode is second-order predation, where Hsapiens begins to reduce
carnivore populations. Like the Three-Herbivore Model this results in a major crash of
herbivore, carnivore and Hsapiens populations. It occurs later than it does in the Three-
Herbivore model. Hsapiens initially establish a larger population than in the Three-
Herbivore Model that then diminishes when the first crash occurs. In the– Four-
164
Fig. 59. Equilibrium mode graph. Four-herbivore model.
165
Fig. 60. Second predator (Overkill), Hsapiens enters the New World. Four-herbivore model.
166
Herbivore Model there is a second dip in herbivore, carnivore and Hsapiens populations
and a second increase in plants after the first crash. Results are shown in Figure 61.
To see why these results were obtained it is useful to look at herbivore populations
shown in Figure 62.
As in the Three-Herbivore Model browsers and mixed feeders expand and crash.
Ruminant grazers boom and non-ruminant grazers slump. Oscillations in the two grazer
populations vary inversely. When browsers and mixed feeders crash ruminant grazers dip
severely as they bear all the pressure of predation from both carnivores and Hsapiens
allowing non-ruminants to boom. As Hsapiens and carnivore populations diminish,
grazers rebound, out competing non- ruminants. To understand the dynamics of the
herbivore populations it is necessary to look at the vegetation shown in Figure 63.
Browsers and mixed feeders eat trees faster than trees can recruit, resulting in a
complete crash of small trees, followed by near extinction of large trees as well. The
competition between the two grazer populations diminishes grass allowing trees to
colonize new territory. The take over by trees drives the non-ruminants to final
extinction.
At values less than 0.02 for the amount H. sapiens kills carnivores, browsers escape
extinction. Mixed feeders go extinct at anything over 0.014. At 0.014 and below there is
no extinction. This is shown in Figure 64
167
Fig. 61. Second-order predation, aggregated view. Four-herbivore model.
168
Fig. 62. Second-order predation, herbivores. Four-herbivore model.
169
Fig. 63. Second-order predation, plants. Four-herbivore model.
170
Fig. 64. A. Second-order predation comparative graphs of A. browsers and B.mixed feeders
C.Aggregate with AmtHsKillCrn=0.015. Four herbivore model
171
Chapter III: Testing and Validity
Introduction
In the previous chapter, I presented the models and the results of each step. In this
chapter, I will discuss the validity of the model. Validity consists of establishing methods
and a rationale for testing a model, as well as judging its legitimacy in relation to its
purpose. According to Richardson and Pugh, “[i]t is meaningless to try to judge validity
in the absence of a clear view of model purpose” (1981, p. 312).
There are two purposes to this modeling effort: first, that the dynamics of the second-
order predation hypothesis be explained; and second, that both the overkill hypothesis
and the second-order predation hypothesis be simulated using the same assumptions and
values. The first purpose is a necessary precondition of the second. Before simulating the
second-order predation hypothesis in the same modeling environment as the second-
predator overkill hypothesis, it is important to know how the former works and what
impacts it would have had if, in fact, it were the cause of megafaunal extinctions. The
second purpose — simulating both hypotheses using the same assumptions and values —
is necessary in order to evaluate the new theory in the same context as the old.
Once the model’s purpose has been determined, it becomes necessary to look at its
suitability and consistency. These validation characteristics occur on two levels: one
focuses on the model’s structure; the other, on its behavior. These characteristics can be
organized into a matrix, as shown in Table 5.
172
Table 5 – Validity matrix based on Richardson and Pugh (1981)
Focusing on Structure Focusing on Behavior
Testing for Suitability
of purpose (tests
focusing inward on the
model)
Dimensional consistency
Extreme conditions in equations
Boundary adequacy of variables
Parameter (in)sensitivity
behavior characteristics
Structural (in)sensitivitybehavior characteristics
Testing for
Consistency of purpose
(tests comparing the
model with the real
system)
Face validity - rates and levels
- information feedback
- delays
Parameter values - conceptual fit
- numerical fit
Replication of reference modes(boundary adequacy for behavior)
- problem behavior
- anticipated behavior
Surprise behavior
Extreme condition behavior
Contributing to Utility
& Effectiveness of a
suitable consistent
model
Appropriateness of model
characteristics for audience
- size
- simplicity/complexity
- aggregation/detail
Counter intuitive behavior
- exhibited by model
- made intuitive by model analysis
Tests for Suitability of Structure
The questions to be asked regarding the suitability of structure involve issues of
dimensional consistency, extreme conditions, and boundary adequacy.
Dimensional Consistency
Do the dimensions of the variables in every equation of the model agree with the
computation? (This question tests for internal, or mathematical, validity.)
In the previous chapter, I presented the equations for all four steps. These equations
are internally consistent, and therefore the models have mathematical validity.
173
Extreme Conditions
Do the equations in the model continue to hold when tested under all possible extremes
for all variables?
For the most part, the equations perform as expected under extreme conditions. When
H. sapiens migrates into the New World in very high numbers, the second-order
predation simulation changes to show that there is less extinction rather than more. The
increase in killing of carnivores is irrelevant to the functioning of the ecosystem, because
the impact of H. sapiens replaces the impact of the non-human carnivores being killed.
The comparison graphs of carnivores, herbivores, plants and H. sapiens are shown in
Figure 65.
The same is true if the model is tested with extreme food values. If H. sapiens requires
the same or more food then a non-human carnivore, then the impact of killing carnivores
is made irrelevant. H. sapiens replaces non-human carnivores from the perspective of
herbivores and plants. The comparison graphs of carnivores, herbivores, plants and H.
sapiens are shown in Figure 66.
If there is no killing of carnivores and if food is varied, then there is no extinction.
These situations are shown in Figure 67.
174
Fig. 65. Comparative populations predicated on varying migration of H. sapiens over time.
175
Fig. 66. Comparative population sizes predicated on varying food needs of H. sapiens over time.
176
Fig. 67. Comparative populations predicated on varying food needs of H. sapiens and an absence
of second-order predation over time.
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Does the model behave reasonably under extreme conditions?
The model scales and behaves as expected under extreme conditions. When run so
that the immigration of H. sapiens is set at 100,000 and all other things are equal, it
shows no boom in herbivore populations and no extinction. At this high immigration
value, the pressure of hunting by H. sapiens replaces the pressure of hunting by
carnivores, so that there is no boom-and-bust pattern and, hence, no extinction. This is
consistent with the thinking behind this model.
When run so that the immigration of H. sapiens is set at 100,000 and H. sapiens food
need set at 10, and the amount H. sapiens kills carnivores set to 0.075 it shows a different
pattern of extinction. Non-ruminant grazers go extinct first followed by mixed feeders.
We said above that non-ruminant grazers respond more to competition with ruminant
grazers than to the reduction in carnivore population. At these values the destruction of
carnivores is so great and so sudden that ruminant grazers experience an immediate boom
that drives the non-ruminant grazers to extinction. The extinction of non-ruminants leaves
more food for mixed feeders. This in turn relieves the pressure on browsers until their
population equilibrates. Mixed feeders eventually are unable to compete with browsers
and ruminant grazers and go extinct. This situation is shown in Figure 68.
Running the model with the immigration of H. sapiens is set at 100,000 and
FoodNeedHs set at 30, results in instant extinction of browsers, ruminant grazers, mixed
feeders and a boom in non-ruminant grazers followed by extinction. That is instantly
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Fig. 68. Herbivore populations where AmtHsMIgrate is set at 100,000, FoodNeedHs is set at 10,
and AmtHsKillCrn=0.075.
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followed by a extinction of H. sapiens and carnivores. Since both the immigration values
and the FoodNeedHs values are extremely unlikely, it does not falsify the hypothesis.
Appendix B shows the varying patterns of extinction and non-extinction when the
model is run at various values.
Boundary Adequacy
Are all the relevant variables and feedback effects necessary? Are all the relevant
variables and feedback effects included?
A basic specification for the model is that it have the fewest possible variables, yet
still be able to achieve its purpose. The base model of Steps 1 and 2 is the simplest
possible ecosystem to have a full complement of players. However, it is too simple to
explain the scenario presented in Chapter I. It does not address the bias toward grazers
and against browsers and mixed feeders, nor does it address the success of ruminant
grazers. By the same token, it cannot address issues of tree and grass usage and survival.
Thus, the three-herbivore and four-herbivore models of Steps 3 and 4 were created out of
insights gleaned from the shortcomings of the base model.
The dynamics of the second-order predation hypothesis can be simulated with the
four-herbivore model, in which all relevant players are accounted for. This scenario holds
that changes in continentality and vegetative patchiness are endogenous, or part of the
system under consideration. During the boom in browsers and mixed feeders, there is a
loss of trees. Trees subsequently repopulate, starting from refugia on the coasts. Because
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they suffer no predation by herbivores, they expand to fill in all the space allotted to
them, stopping only when they reach the boundary of the plains. In Chapter I, above, I
suggested (following (McDonald, 1989 and Reher, 1978) that in the early Holocene, the
boundary between the plains and the eastern woodland was maintained by bison. Thus, in
the second-order predation scenario, the loss of patchiness is due to the boom-and-bust
dynamic.
Continentality is partially a function of vegetation. The loss of tree cover from the
plains would result in a loss of transpired moisture, which would, in turn, bring about an
increase in continentality. This climate change would in itself tend to discourage tree
growth. The model suggests that the initial loss of tree cover may have been due to the
boom-and-bust dynamic.
In the model, trees are constrained by the WoodMix function. After the extinction of
browsers, mixed feeders, and non-ruminant grazers, trees fill in until stopped by the
WoodMix function. It arbitrarily assumes the role of the boundary of the plains. The
model does not create the necessary conditions for that boundary. Thus, only if one
accepts that trees would be limited in the manner specified by the WoodMix function can
one say that the model is valid with respect to the in-filling of trees. In addition, the
model does not address the role of fire in maintaining grasslands. Anthropogenic and
natural outbreak of fire is a feature of Holocene grasslands. Since the model does not
include humidity or fire, it is not valid with respect to the increase in continentality.
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Current hypotheses of extinction do not include all relevant variables. Both the
overkill hypothesis and the various climate-change hypotheses presented in previous
chapters hold continentality and patchiness to be completely exogenous to the model. If
one holds the notion that the world is consistent, then theories other than those related to
extinction are necessary to explain changes in continentality and patchiness. This makes
other theories of extinction needlessly complicated and hence less parsimonious. That
said, the model presented here suggests reasons for a loss of patchiness and a possible
contribution to increased continentality, but it does not create those conditions. One can
see the in-filling of trees up to the boundary specified by the WoodMix function, but
since the model is not geographically based, there is no way of either specifying or
knowing the distribution of trees.
Tests for Suitability of Model Behavior
The questions to be addressed with regard to tests for suitability of behavior deal with
parameter sensitivity and structural sensitivity.
Parameter sensitivity
Is the behavior of the model sensitive to reasonable variations in values?
If the desired behavior is exhibited only within a narrow range of values, then the
model will be held to be too sensitive to be useful. This is especially important when
there are few variables that have values based on observation. Thus, it is important to test
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model behavior over a range of values for each variable and for each combination of
variables.
The specification for this test of validity raises interesting questions for this particular
work. In the general work of modeling, it is important that modelers exhibit desired
behavior over a wide range of values, because they want to know if the dynamics are
generally applicable to the issue under examination. However, if one assumes that
ecosystems, for the most part, tend toward equilibrium and that extinction is a fairly rare
event, then it should be relatively difficult to produce. What one expects to see is a
threshold beyond which extinction occurs Therefore, the specification for this test of
validity should, to some extent, be inverted without being a special case, meaning that
extinction will occur only under fairly narrow conditions.
As mentioned above, at extreme values the model indicates no extinction. From the
first chapter, one learned that animals survived similar climatic change in previous
interglacials. Thus, one knows that extinction is relatively difficult to produce.
Table 6, below, shows that there is a threshold value of 0.02 pounds of carnivores
killed per pound of H. sapiens per year, after which herbivores will become extinct. This
also suggests a motive for H. sapiens to begin a policy of killing predators. Killing
predators at less than the threshold value results in larger populations of H. sapiens and
herbivores. It is only when the killing exceeds 0.02 pounds of carnivores per pound of H.
sapiens per year that extinction occurs.
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Table – 6. – Carnivore population reduction
Carnivore Reduction Final Herbivore
Population
Final H. sapiens
Population
Overkill ---- 0.000 4,606.66 833.79
0.005 4,702.97 850.66
0.010 4,812.18 869.79
0.015 4,882.19 883.50
Extinction --- -0.02 882.87 180.53
Structural Sensitivity
Is the behavior of the model sensitive to reasonable alternative formulations?
The model has been built up incrementally, and each step has been separately tested.
The sensitivity of the model to alternative formulations has been demonstrated. The three
reference modes are alternative formulations of the ecosystem under consideration, and
the models are consistent with their reference modes.
Tests for the Consistency of the Model with the Real system
Tests for the consistency of the model’s structure with the real system address issues of
face validity, parameter values, and the replication of reference modes.
Face Validity
Does the model’s structure look like the real system?
The purpose of the model, which informs its various criteria, is to explain what
happened during the Pleistocene-Holocene transition. This goal suggests that the least
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complex model that still can explain the phenomena should be used. While the base
model partially meets this standard by simulating a very simple ecosystem, it fails in that
it does not produce extinction. It is also unrealistic in assuming a flat percentage
reduction in carnivore populations and not taking into account the survival of ruminants
over non-ruminants and of grazers over browsers. It is also not sufficiently complex to
differentiate between vegetation types. For all these reasons, I elaborated on the base
model in the Step 3 and Step 4 models, although disaggregation was minimal. Therefore,
in its final guise and within the explanatory mandate, the model is valid as regards
consistency with the real system.
Parameter Values
Are the parameters recognizable in terms of the real system? Are the values selected
consistent with the best information we have about the real system?
The parameters I used were based on previous models produced by Mosimann and
Martin (1975) and Whittington and Dyke (1984). Some of the values I selected were
based loosely on the literature and also on conversations with professionals working with
extant animals, as explained in the definitions provided in the previous chapters. I derived
the remaining values from running the model and making assumptions based on the role
of dynamic equilibrium.
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Replication of Reference Modes
Do the models adequately reproduce the reference modes?
The reference mode for the model before the introduction of H. sapiens is one of
dynamic equilibrium. This suggests that in graphing the levels for all stocks over time,
the lines should be flat, but each stock should respond to a disturbance in any of the other
stocks.
In all cases, the model reproduces the dynamic-equilibrium reference mode.
However, because equilibrium was one of the assumptions of the modeling effort, this
replication was to be expected.
The second-predator overkill reference mode is also replicated in all cases.
The second-order predation reference mode is not replicated in the aggregated model.
It produces a boom-and-bust pattern, but as the model is not sufficiently complex to
differentiate among herbivores, no extinction is shown. It merely serves to suggest that a
boom-and-bust pattern would occur if predator populations were reduced. This made it
necessary to create the disaggregated models that do reproduce the reference modes and
patterns that were anticipated by the extinction scenario. The three-herbivore model
produces extinction in browsers and mixed feeders, and the four-herbivore model shows
that ruminant grazers were more competitive than non-ruminant grazers.
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Surprise behavior
Does the model produce behavior that is surprising under some conditions?
Here it is possible that the mechanisms producing the anomalous behavior are real
and the behavior meaningful.
The basic model shows a boom-and-bust pattern, but no extinction; the three-
herbivore model has one extinction event; and the four-herbivore model has two
extinction events.
That there is less extinction when H. sapiens kills carnivores at lower rates may seem
surprising but it suggests reasons for the differences in extinction patterns.
These variations suggests two things: first, that the extinction event is likely to have
been more complex than is generally assumed; and second, that the more complex the
ecosystem, the more complex the extinction event is likely to have been.
Additional Characteristics Contributing to Model Utility and Effectiveness
Several other factors contribute to the utility and effectiveness of a suitably consistent
model, including appropriateness of structure, an explanation of counterintuitive
behavior, and the generation of new insights.
Appropriateness of Structure:
Since the model is meant to explain rather than to prove, its structure should serve this
purpose. Thus, an appropriate structure is one that is simple enough to be understood by
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informed members of the profession, but also one that is rich enough to produce the
desired behavior. The base model is simple enough to be understood, but it does not
produce sufficiently complex behavior. For this reason, I elaborated on the base model in
order to introduce more complexity. Nevertheless, it was necessary to begin with the
simple model, for without it, the more complex models might have been too hard to
understand. Thus, both the basic and disaggregated models are necessary for
understanding the dynamics of the scenario presented in the first chapter.
Counterintuitive Behavior:
The production of counterintuitive behavior is expected, but the model itself should make
the reasons for such behavior clear. For example, the path of extinction proposed by the
second-order predation hypothesis is initially counterintuitive; yet the model plays a role
in clarifying this behavior.
Intuitively, one would expect that the more meat H. sapiens consumed, the sooner
extinction would have occurred. In fact, the model shows that the exact opposite obtains.
The amount of food required by H. sapiens (FoodNeedHs) has a default value of 10
pounds per pound of H. sapiens per year. This is half of what is required by a top
carnivore (lion, tiger, wolf, etc.). If I reduce the amount of food required, the pattern of
extinction occurs sooner. If I increase the amount of food required, the pattern of
extinction occurs later. As is shown above, if the amount of food required is equal to that
required by a top carnivore, then herbivores escape extinction.
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Generation of Insights
I expect the creation and presentation of this model to generate new areas and questions
for research. I will deal with this topic in the Chapter IV, Conclusions and Significance.
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Chapter IV: Conclusions and Significance
Introduction
Like validity, conclusions are meaningless unless they relate to purpose. As I have stated
before, there are two purposes to this work: first, that the dynamics of the second-order
predation hypothesis be explained; and second, that both the overkill hypothesis and the
second-order predation hypothesis be simulated using the same assumptions and values.
Conclusions
In the base model, I addressed the first purpose by showing how the dynamic of boom-
and-bust impacts plants, herbivores, and carnivores. In the elaborations of the base
model, I showed how the same dynamic acts upon trees and grass, and browsers, grazers,
and mixed feeders. In the first chapter I said:
“The task of the model is to explore the hypothesis that second-order predation
resulted in an overpopulation of herbivores which overgrazed their environment
resulting in widespread extinction.”
The second purpose of this work was to simulate both overkill and second-order
predation hypotheses in the same modeling environment and using the same assumptions,
so that they could be evaluated in the same context. The hypothesis to be tested was:
“Second-order predation and its subsequent boom-and- bust ecological dynamic,
explains the data better than overkill alone? The alternative hypothesis is that overkill
alone explains the data better.”
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The criterion for success as stated in Chapter II was as follows:
“I will consider the modeling and simulation project a success if it indicates the
conditions under which the second-order predation hypothesis would have operated.
If these conditions are found to be less likely to produce extinction than those
required for the overkill hypothesis then the second-order predation hypothesis will
be rejected.”
Using the same assumptions and starting values, conditions imposed by the overkill
hypothesis are less likely to produce extinction than conditions imposed by the second-
order predation hypothesis. Put more briefly, second-order predation is more consistent
with extinction than is overkill. Thus, the alternative hypothesis is rejected.
Discussion
The second-order predation, as a working hypothesis, explains features of the
Pleistocene-Holocene transition that were previously unexplained. (Some of these
features are shown in Table 2 of Chapter I, p. 27) The second-order predation hypothesis
allows me to suggest some reasons for several of the differences between the Pleistocene
and the Holocene.
Climate and Vegetation
The second-order predation hypothesis, as a working hypothesis, suggests that during the
Holocene, trees radically declined due to herbivore over-browsing. When trees
repopulated, they would have filled in from the mountain refugia where they had
survived. This in-filling would have created closed-canopy forests, as there would have
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been no large herbivores, like elephants, to knock down trees and establish gaps in the
forest. This sequence of events has been shown to have occurred in recent times in
Australia, when Aboriginal fire-stick farming was eliminated in areas where there were
no large herbivores (Flannery, 1995).
At the same time that trees were filling in from the mountain ranges, bison would
have been repopulating the plains. Giant herds of new, smaller, obligate-grazing bison
would have assumed a different life style in order to cope with scarcity conditions. By
roaming and foraging over vast areas, they would have maintained their grazing lawns
and kept the forest from encroaching (McNaughton, 1979, 1984; McNaughton et al,
1986). At some point, a balance would have been reached, where the plains maintained
by bison and the closed canopy forest would have met. On the plains, because there were
very few trees, there would have been very little transpiration and hence a low relative
humidity and an increase in continentality.
As a working hypothesis, the second-order predation hypothesis, may take us further
in explaining more recent extinction events. As pointed out by Burney and MacPhee:
Although the extinctions in Madagascar occurred only a short time ago, they
resemble those that came at the end of the Pleistocene. (Burney & MacPhee,
1988)
The extinction event in Madagascar coincided with the introduction of people and is
associated with a change in the vegetation. Burney reports:
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A sediment core from Lake Kavitaha, central Madagascar, provides a stratigraphic
record of changes in pollen spectra and charcoal influx in the late Holocene. The
earliest pollen spectra distantly resemble the modern pollen rain of a vegetational
mosaic in northern Madagascar, although results of principal component analysis
suggest no close modern analog. A about 1300 yr. B.P., a marked rise in charcoal is
followed by a decline in pollen of woody taxa, culminating in a change to grass-
dominated pollen spectra within about 4 centuries. Pollen of woody taxa decline
below 15% of total terrestrial pollen and spores beginning about 600 yr. B.P. The
influx of charcoal from graminoid sources remains high until recent centuries. The
late Holocene changes in vegetation and fire ecology at the site were approximately
contemporaneous with the latest 14C dates for the extinct megafauna and the earliest
dates for human occupation. (Burney, 1988)
The model would suggest that the boom phase of the boom/bust pattern accounted for
the change from woody taxa to grass dominated pollen and the increase in aridity. Once a
new equilibrium was reached fire the increase in aridity itself would exacerbate the
tendency for fires, natural and anthropegenic to keep the landscape in grasses.
Animals
I said in the first chapter:
…In order to be an improvement, any new hypothesis must address and explain: 1)
the extinction of horses in North America; 2) the extinction of the ground sloth; 3) the
bias in favor of ruminants; and 4) the bias in favor of small mammal size.
If one accepts the second-order predation hypothesis, as a working hypothesis, all
four of these issues can be explained. The extinction of horses would have been due to
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starvation during the boom-and-bust dynamic. The fact that they currently survive in a
Holocene climate is no longer an anomaly. The period of boom-and-bust was a sort of
ecological bottleneck, through which animals like horses were not able to pass. But when
they were eventually reintroduced in North America, they were able to successfully adapt
to the warmer, more continental climate.
In addition, mixed feeders went extinct as quickly as did browsers and did not
experience as large a boom period. In the model, mixed feeders respond to the boom in
browser and grazer populations more than they respond to the reduction in carnivores.
This suggests reasons why the North American camel, which was a mixed feeder, was
not able to survive the “bottleneck.”
Similar forces at work can also explain the extinction of the ground sloth. Hansen’s
(1978) study of ground sloth dung deposited between 13,000 BP and 10,000 BP found
that over time, the percentage of dietary supplement plants found in the dung declined.
More and more frequently, the sloth’s diet consisted less of its staple food, globemallow,
and more of a plant known as Mormon tea. Mormon tea is shunned by most herbivores,
probably because it contains more deleterious antiherbivory compounds than other plants
with similar protein and energy compositions (Guthrie,1989). At first, the ground sloth to
was able to process these toxins, although Mormon tea was not its preferred food. But
slowly the percentage of Mormon tea surpassed the staple, which was becoming
increasingly scarce. Thus, one sees a pattern of ever-increasing consumption of an
undesirable food, until the sloth finally disappeared. If one accepts the second-order
predation hypothesis, this is a picture of an animal that was increasingly unable to find its
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preferred food. Initially, it simply ate more and more of a food it could tolerate, but one
that was not eaten by other animals. Over time, the less desirable food came to dominate
the sloth’s entire diet. Ultimately, the sloth was unable to process a steady diet of
antiherbivory toxins and died out as a result (Phillips, 1988).
Although the model is a vast simplification of the real world, accepting the second-
order predation hypothesis makes the bias in favor of ruminants and smaller size animals
comprehensible. In a condition of environmental exhaustion, animals that are more
efficient will be favored. All things being equal, ruminants will extract more from the
same quantity of forage than non-ruminants. Similarly, within a species, smaller animals
will be competitively favored because they require less food to attain reproductive
maturity and will therefore leave more offspring.
Thus, the new hypothesis of second-order predation explains such observations better
than do the existing hypotheses.
Archaeological Evidence
Applying the implications of second-order predation to a perspective on emerging human
populations, one might consider the phenomenon of Paleo-Indian big-game hunting as a
response to the overpopulation of herbivores. Yellen and Harpending (1972), Yellen
(1976), and Harpending and Davis (1976) suggest that an environment in which
resources are widely scattered and only locally abundant tends to support group mobility
and a networked social structure. During the boom phase or the boom-and-bust cycle,
people would have found gathering their food more difficult because of the scarcity of
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plants. Scavenging for or hunting animals would have been easier because it would have
taken them less time to locate an overabundant animal population. If, as suggested by
Haynes (1995), many animals were starving because of a lack of forage, it would have
made sense to scavenge for dead ones and kill off weakened ones. The more complex of
my models suggests that there would have been an abundance of one kind of herbivore
while another kind was going extinct. This would have exacerbated local abundance and
local scarcity and would have made a networked, highly mobile lifestyle more likely.
Following this line of reasoning, the homogeneity of Paleo-Indian artifacts can be
explained in two ways. First, people who did not adopt the mobile, networked lifestyle
that maximized the use of weakened and dying animals would have had a hard time
finding sufficient resources. Second, those people who followed herbivores would have
met and exchanged information, they probably also exchanged cultural practices, which
resulted in a certain uniformity of artifacts.
With the extinction of megafauna, two changes would have occurred. The first would
have been a crash in the size of human populations that continued to rely mainly on
hunting and scavenging. The second change would have taken place in the lifestyle of
those that managed to survive the extinction event. Living in a new ecosystem with a
depauperate fauna and re-established vegetation, these people would have turned to
gathering plants for their food and consequently would have adopted a more sedentary
lifestyle (Yellen and Harpending, 1972; Yellen, 1976; and Harpending and Davis, 1976).
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With the increase in reliable resources, the world view of the survivors might also
have changed. They would have seen or heard about the massive starvation and therefore
might have been hesitant to share anything that they saw as necessary to the survival of
their own small group. This change in perspective might have resulted in a kind of Paleo-
Dark Ages. As in any dark age, knowledge and skills would have been lost. Such a
scenario would account for the drop in the quality of artifacts, as well as the loss of
homogeneity.
If one assumes that the second-order predation hypothesis holds more generally, then
the rest of the Pleistocene needs to be re-examined in light of this assumption. It may
have implications for the invention of agriculture and for the invention of war (Whitney-
Smith, 1995).
Implications for Further Research
The first challenge for further research would be to find field evidence to support or
falsify the theory. Dating evidence, which might show that the extinctions took place
over a longer period of time, would throw serious doubt on the hypothesis. It might also
be falsified if the synchronicity of extinctions were found to be in error.
Evidence that H. sapiens hunted carnivores in the New World would be helpful in
supporting the theory. According to Price (1986), remains of wild cats have been found in
late Pleistocene sites in Belgium. Soffer (1985) suggests that the large number of foot
bones of fur-bearing animals found in late Pleistocene archaeological sites in Siberia
show that humans hunted carnivores for their fur.
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More implications for further research are rooted in the shortcomings pointed out in
the validity chapter. The first would be to model a larger system that would include the
feedback loop between vegetation and relative humidity. To do this, it would be
necessary to determine the transpiration rate and moisture requirements of a mixed
parkland. With this information, one could create a feedback between the two tree stocks
and relative humidity that would be tied back to tree production.
Another research project would be to create a model of the Pleistocene, so that the
dynamics could work within geographic cells. Such a model could test whether or not
trees would behave as the hypothesis has suggested.
Yet another project would be to model the impact of fire on the plant-herbivore
system. Flannery (1995) suggests that the use of fire by H. sapiens is a result of the
extinction of large herbivores rather than a cause. He argues that once the large
herbivores of Australia were extinct, Aboriginal people used fire to keep the landscape
open. When Europeans excluded the small, anthropogenic fires, the landscape filled in,
and the newly created closed-canopy forests, with their dense understoreys, led to larger,
more disastrous fires. Thus, the model could be used to simulate fire before, during, and
after the extinction of herbivores. The results of this modeling effort could then be
compared with what exists in the archaeological record.
Significance of the Model for Science and Research
The significance of this model is that it makes an attempt to specify a mechanism
whereby climate and the introduction of H. sapiens could have combined to result in
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megafaunal extinction during the Pleistocene-Holocene transition. Furthermore, the
modeling environment that has been created for this project can be adapted quite easily to
simulate other hypotheses of extinction.
The models created for this project have been given interfaces to allow users to
specify the values they want for many variables. In the interfaces, some variables are
controlled by sliders, as shown in Figure 69A. Moving slider one (1) lets users change the
number of H. sapiens entering the New World. The number of carnivores killed by H.
sapiens is changed by moving slider two (2). Slider three (3) controls the number of
pounds of herbivores required to support a pound of H. sapiens per year. The
EquilibriumSwitch runs the model in dynamic equilibrium mode. It inserts a one-time
pulse reduction in carnivore population, which shows up in all the other sectors.
By changing the slopes of the various graphs, as shown in Figure 69B, users can
change the hunting rates for carnivores or H. sapiens, the birth and efficiency rates of the
various herbivores, and the limit allocated to trees.
Thus, with the help of the interfaces, students of the Pleistocene can use the model as
an experimental space. When they input their own values, the dynamics will become
clearer, and it will be easier for them to see that the result is due less to specific values
than to the dynamics of the system itself.
The other implication for research is the use of a model as an exploratory device to
test the consistency of one hypothesis against another. By creating a standard in which
hypotheses can be tested using the same assumptions and starting values, models can
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Fig. 69. Interface to the model:A, using the slider; and B, changing the curve
A.
.B
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become laboratories. They will prove to be useful to archaeologists, paleontologists,
paleo-ecologists, and any other scientists in fields where field observations are likely to
yield incomplete evidence. By using models, they will be able to test which hypotheses
are most consistent with the observations that are possible.
A Broader Significance for the Modern World
The model shows that extinctions are more likely to occur because of overpopulation and
the resultant destruction of habitat than through overhunting. It suggests that we need to
be careful ourselves that we do not so overpopulate the earth that we destroy our own
habitat.
We hear every day in the news that more and more habitats are being destroyed and
more and more species are being threatened. Are the endangered species, like canaries in
the mine shaft, warning us that we are also on the brink of another great extinction event,
one that could have as dire an implication for us as conditions at the end of the
Pleistocene had for the megaherbivores? The frightening thing is that just as we do not
know what the threshold of destruction was in ages past, we also do not know what the
future threshold of our own destruction might be. Nor do we know whether we might be
able, through chance or intention, to pull back from it.
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Appendix A: Equations
This appendix lists all the equations in the final, four-herbivore model. It is presented to
make replication of the model easier.
Continent
Area = 30000
GrassK = K*(1-WoodMix)
K = Area*Kmult
Kmult = 25
Plants = BigTrees + SmallTrees + GrassHigh + GrassLow
TreeK = K*WoodMix
TreePrC = 1-AllTrees/Plants
WoodMix = GRAPH(TreePrC)
(0.4, 0.697), (0.46, 0.438), (0.52, 0.354), (0.58, 0.319), (0.64, 0.301), (0.7, 0.27),
(0.76, 0.249), (0.82, 0.231), (0.88, 0.196), (0.94, 0.144), (1.00, 0.00)DOCUMENT: Controller for the amount of carrying capacity allotted to trees. The rest is
allocated to grass.
Trees
BigTrees(t) = BigTrees(t - dt) + (Maturity - OutBt) * dt
INIT BigTrees = 93584.5900DOCUMENT: The stock of Big Trees (measured in a.u.(animal units))
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Maturity = SmallTrees*MatRate
OutBt = BtEating+(BtDeath*BigTrees)
SmallTrees(t) = SmallTrees(t - dt) + (InTrees - Maturity - OutSt) * dt
INIT SmallTrees = 18716.9200
DOCUMENT: The Stock of Small Trees (measured in a.u.(animal units))
InTrees = If AllTrees<.001 then Reseed else TreeRepo*(TreeRepoRate*(1-
AllTrees/TreeK))
Maturity = SmallTrees*MatRate
OutSt = StEating+(SmallTrees*StDeath)
AllTrees = BigTrees + SmallTrees
AmtEatBt = (BzMxTreeNeed-SmallTrees)*BtRatio
BtDeath = .03
BtEating = If BtNeed > BigTrees then BigTrees else BtNeed
BtNeed = If BzMxTreeNeed > SmallTrees then AmtEatBt else 0
BtRatio = 1.5
BtRepo = BigTrees*.4
BzMxTreeNeed = TreesNeedBz + TreesNeedMx
MatRate = .15
203
Reseed = 1
StDeath = .05
StEating = If BzMxTreeNeed>SmallTrees then SmallTrees else BzMxTreeNeed
StRepo = SmallTrees*.6
TreeRepo = BtRepo+StRepo
TreeRepoRate = if SmallTrees=0 then 0 else .25
Grass
GrassHigh(t) = GrassHigh(t - dt) + (InHi - OutHi - HitoLo) * dt
INIT GrassHigh = 25624.7500DOCUMENT: The stock of high quality grass (measured in a.u.(animal units))
InHi = GrRepo*(RateGr*(1-AllGrass/GrassK))
OutHi = If GrassHigh > GrassNeed then GrassNeed + (GrassHigh*DRateH) else
GrassHigh
HitoLo = GrassHigh*HiLoRate
GrassLow(t) = GrassLow(t - dt) + (HitoLo - OutLo) * dt
INIT GrassLow = 169123.3600DOCUMENT: The stock of low quality grass (measured in a.u.(animal units))
HitoLo = GrassHigh*HiLoRate
OutLo = (GrassLow*DRateLo)+Eatlo
204
AllGrass = GrassLow + GrassHigh
AmtLo = (GrasNeedGz[Ruminant]-AvailHi[Ruminant])*LoGRatio
AvailHi[Ruminant] = GrassHigh*RatiosGz[Ruminant]
AvailHi[NonRuminant] = GrassHigh*RatiosGz[NonRuminant]
DRateH = .01
DRateLo = .1
Eatlo = If GrassNeed >GrassHigh THEN AmtLo ELSE 0
GrassNeed = ARRAYSUM(GrasNeedGz[*]) + GrassNeedMx
GrRepo = HiRepo+LoRepo
HiLoRate = .66
HiRepo = GrassHigh*.5
LoGRatio = 1.5
LoRepo = GrassLow*.5
RateGr = .33
Browsers
Browsers(t) = Browsers(t - dt) + (InBz - OutBz) * dt
INIT Browsers = 1619.180DOCUMENT: The stock of browsers n thousand pounds
205
InBz = Browsers*(BzBirth*(1-Browsers/AllTrees))
OutBz = Browsers*(DeathRateBz)+HuntBz
BzBirth = BzBirthRate*BzDesnityEffectGrf
BzBirthRate = .6
BzEfficiency = 1
DeathRateBz = .4
HuntBz = delay((BzHHs+BzHCrn),HuntDelay)
TreesNeedBz = Browsers*BzEfficiency
BzDesnityEffectGrf = GRAPH(BzDensity)
(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.28),
(0.0136, 0.825), (0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00),
(0.0273, 1.00), (0.03, 1.00)
Grazers
Grazers[Ruminant](t) = Grazers[Ruminant](t - dt) + (InGz[Ruminant] -
OutGz[Ruminant]) * dt
INIT Grazers[Ruminant] = 1043.130DOCUMENT: The stock of ruminant grazers in thousand pounds
Grazers[NonRuminant](t) = Grazers[NonRuminant](t - dt) + (InGz[NonRuminant] -
OutGz[NonRuminant]) * dt
206
INIT Grazers[NonRuminant] = 747.750
DOCUMENT: The stock of non-ruminant grazers in thousand pounds
InGz[Ruminant] =
Grazers[Ruminant]*(GzBirth[Ruminant]*(1-Grazers[Ruminant]/AllGrass))
InGz[NonRuminant] =
Grazers[NonRuminant]*(GzBirth[NonRuminant]*(1-Grazers[NonRuminant]/AllGrass))
OutGz[Digestion] = Grazers[Digestion]*(DeathRateGz[Digestion])+HuntGz[Digestion]
DeathRateGz[Digestion] = .4
FoodNeed = (Grazers[Ruminant]* SetEffGz[Ruminant])+(Grazers[NonRuminant]*
SetEffGz[NonRuminant])
GrasNeedGz[Digestion] = Grazers[Digestion]*GzEfficiency[Digestion]
GzBirth[Digestion] = GzDensityEffectGrf[Digestion]*GzBirthGrf[Digestion]
GzBirthGrf[Digestion] = GRAPH(GzEfficiency[Digestion])
GzBirthGrf [Ruminants]
(0.600, 0.591); (0.690, 0.569); (0.780, 0.559); (0.870, 0.551); (0.960, 0.548);
(1.050, 0.545); (1.140, 0.539); (1.230, 0.535); (1.320, 0.528); (1.410, 0.521);
(1.500, 0.508)
GzBirthGrf [NonRuminants]
(0.825, 0.906), (0.900, 0.816), (0.975, 0.730), (1.050, 0.640), (1.250, 0.550),
(1.200, 0.460), (1.275, 0.370), (1.350, 0.276), (1.425, 0.190), (1.500, 0.100 )DOCUMENT: The curve describing the birth rate of Ruminants is flatter than that of
Non-Ruminants. Ruminants are able to withstand a greater degree of environmental
stress.
207
GzEffGrf[Digestion] = GRAPH(GzGrass)
GzEffGrf[Ruminant]
(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, 1.190);
(0.995, 1.010); (0.996, 0.853); (0.997, 0.725); (0.998, 0.620); (0.999, 0.538);
(1.000, 0.515)
GzEffGrf[NonRuminant]
(0.990, 2.000); (0.991, 1.603); (0.992, 1.355); (0.993, 1.168); (0.994, 1.025);
(0.995, 0.890); (0.996, 0.785); (0.997, 0.688); (0.998, 0.613); (0.999, 0.560);
(1.000, 0.523)DOCUMENT: Efficiency curve for NonRuminants is more concave than that of
Ruminants. The curve for Ruminants is more sigmoid. Ruminants are assumed to be
better buffered from fluctuations in the environment than NonRuminants.
GzEfficiency[Digestion] = SetEffGz[Digestion]*GzEffGrf[Digestion]
GzGrass = If AllGrass< 1 then 0 else 1-FoodNeed/AllGrass
HuntDelay = 1.5
HuntGz[Digestion] = Delay((GzHHs[Digestion]+GzHCrn[Digestion]),HuntDelay)
RatiosGz[Digestion] = Grazers[Digestion]/SumGz
SetEffGz[Ruminant] = .8
SetEffGz[NonRuminant] = .9
SumGz = ARRAYSUM(Grazers[*])
208
GzBirthGrf[Digestion] = GRAPH(GzEfficiency[Digestion])
GzBirthGrf[Ruminant] (0.600, 0.591); (0.690, 0.569); (0.780, 0.559); (0.870,
0.551); (0.960, 0.548); (1.050, 0.545); (1.140, 0.539); (1.230, 0.535); (1.320,
0.528); (1.410, 0.521); (1.500, 0.508)
GzBirthGrf[NonRuminant] (0.825, 0.906), (0.900, 0.816), (0.975, 0.730), (1.050,
0.640), (1.250, 0.550), (1.200, 0.460), (1.275, 0.370), (1.350, 0.276), (1.425,
0.190), (1.500, 0.100 )
DOCUMENT: The curve describing the birth rate of Ruminants is flatter than that of
Non-Ruminants. Ruminants are able to withstand a greater degree of environmental
stress.
GzDensityEffectGrf[Digestion] = GRAPH(GzDensity[Digestion])
(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.28),
(0.0136, 0.825), (0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00),
(0.0273, 1.00), (0.03, 1.00)
GzEffGrf[Digestion] = GRAPH(GzGrass)
GzEffGrf[Ruminant]
(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, 1.190);
(0.995, 1.010); (0.996, 0.853); (0.997, 0.725); (0.998, 0.620); (0.999, 0.538);
(1.000, 0.515)
GzEffGrf[NonRuminant]
(0.990, 2.000); (0.991, 1.603); (0.992, 1.355); (0.993, 1.168); (0.994, 1.025);
(0.995, 0.890); (0.996, 0.785); (0.997, 0.688); (0.998, 0.613); (0.999, 0.560);
(1.000, 0.523)
209
DOCUMENT: Efficiency curve for NonRuminants is more concave than that of
Ruminants. The curve for Ruminants is more sigmoid. Ruminants are assumed to be
better buffered from fluctuations in the environment than NonRuminants.
MixedFeeders
MixedFeeders(t) = MixedFeeders(t - dt) + (InMx - OutMx) * dt
INIT MixedFeeders = 1405.070
DOCUMENT: The stock of mixed feeders in thousand pounds
InMx = MixedFeeders*MxBirth
OutMx = MixedFeeders*(MxDeathGrf)+HuntMx
GrassFactorMx = If AllGrass< 1 then 0 else 1- GrassNeedMx/AllGrass
GrassNeedMx = MixedFeeders*(MxEfficiency*(1-MxTreePerC))
HuntMx = delay((MxHHs+MxHCrn),HuntDelay)
MxBirth = MxBirthGrf*MxDensityEffectGrf
MxTreePerC = .5
TreeFactorMx = If AllTrees< 10 then 0 else 1-TreesNeedMx/AllTrees
TreeRatio = If AllTrees < 1 then 0 else AllTrees/(AllTrees+AllGrass)
TreesNeedMx = MixedFeeders*(MxEfficiency*MxTreePerC)
210
MxBirthGrf = GRAPH(TreeFactorMx)
(0.945, 0.00), (0.95, 0.00), (0.955, 0.07), (0.96, 0.137), (0.965, 0.207), (0.97,
0.277), (0.975, 0.347), (0.98, 0.417), (0.985, 0.483), (0.99, 0.557), (0.995, 0.626),
(1.00, 0.697), (1.00, 0.697)DOCUMENT: As trees increase birth rates increase.
MxDeathGrf = GRAPH(GrassFactorMx)
(0.989, 1.00), (0.99, 1.00), (0.991, 0.915), (0.992, 0.825), (0.993, 0.73), (0.994,
0.62), (0.995, 0.525), (0.996, 0.43), (0.997, 0.32), (0.998, 0.225), (0.999, 0.11),
(1.00, 0.015), (1.00, 0.015)
DOCUMENT: As grass increases death decreases.
MxDensityEffectGrf = GRAPH(MxDensity)
(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.28),
(0.0136, 0.825), (0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00),
(0.0273, 1.00), (0.03, 1.00)
MxEfficiency = GRAPH(TreeRatio)
(-0.1, 4.00), (1.39e-017, 4.00), (0.1, 3.70), (0.2, 2.21), (0.3, 1.25), (0.4, 1.05), (0.5,
1.00), (0.6, 1.05), (0.7, 1.25), (0.8, 2.21), (0.9, 3.67), (1.00, 4.00), (1.10, 4.00)
DOCUMENT: As mixed feeders move away from their optimum mix of grass and browse
they become less efficient. The optimum efficiency is at the base of the 'U'.
Carnivores
Carnivores(t) = Carnivores(t - dt) + (InCrn - OutCrn - OutStepCrn) * dt
INIT Carnivores = 842.650DOCUMENT: The stock of carnivores in thousand pounds
InCrn = Carnivores*(BRateCrn*(1-Carnivores/CarnivoreK))DOCUMENT: Intflow to the Carnivore stock.
211
OutCrn = Carnivores*(DRateCrn)
DOCUMENT: Outflow from the Carnivore stock.
OutStepCrn = If EquilibriumTest = 1 then PULSE((Carnivores*0.2), -11000, 0) else
Hsapiens*(CrnDensityGrf*AmtHsKillCrn)DOCUMENT: Secondary outflow from the carnivore stock.
AmtHsKillCrn = 0.025DOCUMENT: The rate at which H. sapiens kills predators.
BRateCrn = .4DOCUMENT: Assigned death rate of carnivores based loosely on wolf data.
CarnivoreK = DELAY(Herbivores,HerbivoreDelay)
DRateCrn = .33
DOCUMENT: Assigned death rate of carnivores based loosely on wolf data.
EquilibriumTest = 0DOCUMENT: Switch on (1) to demonstrate dynamic equilibrium, with a 0.2 one-time
reduction in Carnivores at -11000 years. Overrides settings for AmtMigrateHs and
AmtHsKillCrn.
HerbivoreDelay = 1
CrnDensityGrf = GRAPH(CarnivoreDensity)
(0.00, 0.00), (0.00526, 0.00), (0.0105, 0.105), (0.0158, 0.435), (0.0211, 0.775),
(0.0263, 0.94), (0.0316, 1.00), (0.0368, 1.00), (0.0421, 1.00), (0.0474, 1.00),
(0.0526, 1.00), (0.0579, 1.00), (0.0632, 1.00), (0.0684, 1.00), (0.0737, 1.00),
(0.0789, 1.00), (0.0842, 1.00), (0.0895, 1.00), (0.0947, 1.00), (0.1, 1.00)
Hsapiens
212
Hsapiens(t) = Hsapiens(t - dt) + (InHs + HsMigration - OutHs) * dt
INIT Hsapiens = 0DOCUMENT: The stock of carnivores in thousand pounds. It is initialized at 0 because
the model starts before the migration of Hsapiens to the new world.
InHs = If Hsapiens=0 then 0 else Hsapiens*(BrateHs*(1-Hsapiens/Herbivores))
HsMigration = If EquilibriumTest = 1 then 0 else STEP((AmtMigrateHs),TimeMigrateHs)DOCUMENT: Input to Hsapiens stock. The amount of input is AmtMigrateHs. The time
of input is TimeMigrateHs
OutHs = Hsapiens*(DRateHs)DOCUMENT: Outflow from the stock of Hsapiens.
AmtMigrateHs = .2DOCUMENT: Initial H. sapiens population in thousands.
BrateHs = .04DOCUMENT: Birth rate of Hsapiens as per Whittington & Dyke (1984) modified by an
adjustment for the death rate.
DRateHs = .033DOCUMENT: Death rate of Hsapiens.
TimeMigrateHs = -11500DOCUMENT: The time Hsapiens enters the New World
Density
DOCUMENT: Equations controlling hunting gathered together for ease in programming.
BzDensity = Browsers/Area
BzHCrn = BzHuntingCrn*PrCCrnBz
213
BzHHs = BzHuntingHs*PrCHsBz
BzHuntingCrn = BzHuntGrfCrn*FoodNeedCrn
BzHuntingHs = FoodNeedHs*BzHuntGrfHs
CarnivoreDensity = Carnivores/Area
FoodNeedCrn = 20DOCUMENT: The amount of food needed per year to support one pound of Carnivore. A
50 pound Carnivore would need 50 *20 or 1000 pounds per year.
FoodNeedHs = 10DOCUMENT: The amount of food needed per year to support one pound of H. sapiens. A
100 pound H. sapiens would need 100 *10 or 1000 pounds per year.
GzDensity[Digestion] = Grazers[Digestion]/Area
GzHCrn[Digestion] = GzHuntingCrn[Digestion]*PrCCrnGz[Digestion]
GzHHs[Digestion] = GzHuntingHs[Digestion]*PrCHsGz[Digestion]
GzHuntGrfHs[Digestion] = GRAPH (GzDensity[Digestion])
(0, 0); (0.03, 0.008); (0.06, 0.021); (0.09, 0.039); (0.12, 0.062); (0.15, 0.095);
(0.18, 0.127); (0.21, 0.159); (0.24, 0.183); (0.27, 0.193); (0.3, 0.199)
DOCUMENT: Controls the rate at which Hsapiens hunt Grazers.
GzHuntingCrn[Digestion] = FoodNeedCrn*GzHuntGrfCrn[Digestion]
GzHuntingHs[Digestion] = GzHuntGrfHs[Digestion]*FoodNeedHs
Herbivores = Browsers + MixedFeeders + ARRAYSUM(Grazers[*])
MxDensity = MixedFeeders/Area
214
MxHCrn = MxHuntingCrn*PrCCrnMx
MxHHs = MxHuntingHs*PrCHsMx
MxHuntingCrn = FoodNeedCrn*MxHuntGrfCrn
MxHuntingHs = MxHuntGrfHs*FoodNeedHs
PrCBz = Browsers/Herbivores
PrCCrnBz = Carnivores*PrCBz
PrCCrnGz[Digestion] = Carnivores*PrCGz[Digestion]
PrCCrnMx = Carnivores*PrCMx
PrCGz[Digestion] = Grazers[Digestion]/Herbivores
PrCHsBz = Hsapiens*PrCBz
PrCHsGz[Digestion] = Hsapiens*PrCGz[Digestion]
PrCHsMx = Hsapiens*PrCMx
PrCMx = MixedFeeders/Herbivores
BzHuntGrfCrn = GRAPH(BzDensity)
(-0.02, 0.00), (0.00, 0.00), (0.02, 0.02), (0.04, 0.04), (0.06, 0.061), (0.08, 0.081),
(0.1, 0.099), (0.12, 0.12), (0.14, 0.139), (0.16, 0.16), (0.18, 0.178), (0.2, 0.196),
(0.22, 0.199)
DOCUMENT: Controls the rate at which Carnivores hunt Browsers.
215
BzHuntGrfHs = GRAPH(BzDensity)
(0.00, 0.00), (0.03, 0.003), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15,
0.095), (0.18, 0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.195), (0.3, 0.199)
DOCUMENT: Controls the rate at which Hsapiens hunt Browsers.
GzHuntGrfCrn[Digestion] = GRAPH(GzDensity[Digestion])
(-0.02, 0.00), (0.00, 0.00), (0.02, 0.02), (0.04, 0.04), (0.06, 0.061), (0.08, 0.081),
(0.1, 0.099), (0.12, 0.12), (0.14, 0.139), (0.16, 0.16), (0.18, 0.178), (0.2, 0.196),
(0.22, 0.199)
DOCUMENT: Controls the rate at which Carnivores hunt Grazers.
GzHuntGrfHs[Digestion] = GRAPH(GzDensity[Digestion])
(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095),
(0.18, 0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)
DOCUMENT: Controls the rate at which Hsapiens hunt Grazers.
MxHuntGrfCrn = GRAPH(MxDensity)
(-0.02, 0.00), (0.00, 0.00), (0.02, 0.02), (0.04, 0.04), (0.06, 0.061), (0.08, 0.081),
(0.1, 0.099), (0.12, 0.12), (0.14, 0.139), (0.16, 0.16), (0.18, 0.178), (0.2, 0.196),
(0.22, 0.199)
DOCUMENT: Controls the rate at which Carnivores hunt Mixed Feeders.
MxHuntGrfHs = GRAPH(MxDensity)
(0.00, 0.00), (0.03, 0.003), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15,
0.095), (0.18, 0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.195), (0.3, 0.199)
DOCUMENT: Controls the rate at which Hsapiens hunt Mixed Feeders.
216
Normalization Sector
DOCUMENT: Graphs are shown normalized to 100. This sector is where the
multiplication takes place.
BigTreesNormalized = BigTrees*normBT
BrowsersNormalized = Browsers*normBz
CarnivoresNormalized = Carnivores*normCandHs
GrassHighNormalized = GrassHigh*normGH
GrassLowNormalized = GrassLow*normGL
GrazersNormalized[Ruminant] = Grazers[Ruminant]*normGz[Ruminant]
GrazersNormalized[NonRuminant] = Grazers[NonRuminant]* normGz[NonRuminant]
HerbivoresNormalized = Herbivores * normHrb
HsapiensNormalized = Hsapiens*normCandHs
MixedFeedersNormalized = MixedFeeders* normMx
normBT = 0.001068552
normBz = 0.061759656
normCandHs = 0.118673233
normGH = 0.003902477
normGL = 0.000591284
normGz[Ruminant] = 0.095865328
217
normGz[NonRuminant] = 0.133734537
normHrb = 0.020767871
normMx = 0.071170831
normPl = 0.00032568
normST = 0.005342759
PlantsNormalized = Plants*normPl
SmallTreesNormalized = SmallTrees*normST
218
Appendix B - Summary Graphs
Running the model at various values produces different results. Graphs of the various
runs are below. Graphs are arranged in the following manner. The graphs for various
migration values are on each page: Thus, the graphs on page 1 are all graphs where the
migration of H. sapiens is set at 0.2K, on page 2 H. sapiens migration is set at 10K and
page 3 H. sapiens migration is set at 100K. The columns of graphs have the same value
for FoodNeedHs. The graphs in the first column all have FoodNeedHs set to 10 pounds
of herbivore per year per pound of H. sapiens, second column graphs all have
FoodNeedHs set to 20 pounds of herbivore per year per pound of H. sapiens, second
column graphs all have FoodNeedHs set to 40 pounds of herbivore per year per pound of
H. sapiens. The rows on each page all have the AmtHsKillCrn set to the same value. The
first row on each page AmtHsKillCrn is set to 0 (the position of the overkill hypothesis),
in the second row AmtHsKillCrn is set to 0.01, in the third row AmtHsKillCrn is set to
0.02, in the second row AmtHsKillCrn is set to 0.07. The key to the graphs is below.
0.2 10 0KFM
xtmema37
0.2 20 0KFM
xtmema38
0.2 40 0KFM
xtmema39
0.2 10 0.01KFM
xtmema42
0.2 20 0.01KFM
xtmema43
0.2 40 0.01KFM
xtmema44
0.2 10 0.02KFM
xtmema52
0.2 20 0.02KFM
xtmema53
0.2 40 0.02KFM
xtmema54
0.2 10 0.07KFM
xtii17
0.2 20 0.07KFM
xtii18
0.2 40 0.07KFM
xtii19
219
H. sapiens migrates into the New World 0.2K
10 10 0KFM
xtmemc37
10 20 0KFM
xtmemc38
10 40 0KFM
xtmemc39
10 10 0.01KFM
xtmemc42
10 20 0.01KFM
xtmemc43
10 40 0.01KFM
xtmemc44
10 10 0.02KFM
xtmemc52
10 20 0.02KFM
xtmemc53
10 40 0.02KFM
xtmemc54
10 10 0.07KFM
xtii57
10 20 0.07KFM
xtii58
10 40 0.07KFM
xtii59
220
H. sapiens migrates into the New World 10 K
100 10 0KFM
xtii62
100 20 0KFM
xtii63
100 40 0KFM
xtii64
100 10 0.01KFM
xtii67
100 20 0.01KFM
xtii68
100 40 0.01KFM
xtii69
100 10 0.02KFM
xtii72
100 20 0.02KFM
xtii73
100 40 0.02KFM
xtii74
100 10 0.07KFM
xtii87
100 20 0.07KFM
xtii88
100 40 0.07KFM
xtii89
221
H. sapiens migrates into the New World 100 K
222
Appendix C – Model and Runtime Software on CD Rom
The envelope below contains a CD rom with the four herbivore model and runtime
software. It is formatted for a Windows or Windows NT machine. It also contains the
text of the work in PDF format for printing.
All the above can also be downloaded from my website at
http://quaternary.net/xtinct2000
223
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